Cohomology of group number 851 of order 128

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

General information on the group

  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t6  +  t3  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-5,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring generators

The cohomology ring has 20 minimal generators of maximal degree 9:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. c_1_2, a Duflot regular element of degree 1
  4. b_2_4, an element of degree 2
  5. b_2_5, an element of degree 2
  6. b_3_8, an element of degree 3
  7. b_3_9, an element of degree 3
  8. b_3_10, an element of degree 3
  9. b_4_15, an element of degree 4
  10. b_4_17, an element of degree 4
  11. a_5_21, a nilpotent element of degree 5
  12. b_5_24, an element of degree 5
  13. b_5_25, an element of degree 5
  14. b_6_37, an element of degree 6
  15. b_6_38, an element of degree 6
  16. b_7_52, an element of degree 7
  17. b_7_53, an element of degree 7
  18. b_8_68, an element of degree 8
  19. c_8_72, a Duflot regular element of degree 8
  20. b_9_95, an element of degree 9

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring relations

There are 134 minimal relations of maximal degree 18:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_13
  4. b_2_4·a_1_1
  5. b_2_5·a_1_0
  6. b_2_4·b_2_5
  7. a_1_0·b_3_8
  8. a_1_1·b_3_9 + a_1_1·b_3_8
  9. a_1_0·b_3_9
  10. a_1_0·b_3_10 + b_2_5·a_1_12
  11. a_1_12·b_3_8
  12. b_2_4·b_3_8
  13. b_2_5·b_3_9 + b_2_5·b_3_8 + b_2_52·a_1_1
  14. a_1_12·b_3_10
  15. b_2_4·b_3_10 + b_2_4·b_3_9
  16. b_4_15·a_1_1
  17. b_4_15·a_1_0
  18. b_4_17·a_1_0
  19. b_3_8·b_3_9 + b_3_82 + b_2_5·a_1_1·b_3_8
  20. b_3_9·b_3_10 + b_3_92 + b_3_8·b_3_10 + b_2_53 + b_2_5·a_1_1·b_3_10 + b_2_5·a_1_1·b_3_8
  21. b_2_5·b_4_15
  22. b_3_82 + b_2_53 + b_2_5·a_1_1·b_3_8 + b_4_17·a_1_12
  23. b_3_92 + b_3_82 + b_2_4·b_4_17
  24. b_3_82 + b_2_53 + b_2_5·a_1_1·b_3_8 + a_1_1·a_5_21
  25. a_1_0·a_5_21
  26. b_3_82 + b_2_53 + a_1_1·b_5_24 + b_2_5·a_1_1·b_3_8
  27. a_1_0·b_5_24
  28. b_3_102 + b_3_92 + b_3_82 + b_2_5·b_4_17 + a_1_1·b_5_25
  29. a_1_0·b_5_25
  30. b_4_15·b_3_8
  31. b_4_15·b_3_10 + b_4_15·b_3_9
  32. a_1_1·b_3_8·b_3_10 + b_2_5·a_5_21 + b_2_5·b_4_17·a_1_1
  33. b_2_5·b_5_24 + b_2_5·b_4_17·a_1_1 + b_2_53·a_1_1
  34. b_4_15·b_3_10 + b_2_4·b_5_24 + b_2_4·a_5_21
  35. a_1_12·b_5_25
  36. b_2_4·b_5_25
  37. a_1_1·b_3_8·b_3_10 + b_6_37·a_1_1 + b_2_53·a_1_1
  38. b_6_37·a_1_0 + b_2_4·a_5_21
  39. b_4_17·b_3_8 + b_2_5·b_5_25 + a_1_1·b_3_8·b_3_10 + b_6_38·a_1_1 + b_2_5·b_4_17·a_1_1
  40. b_6_38·a_1_0
  41. b_4_152 + b_2_42·b_4_17
  42. b_3_9·a_5_21 + b_3_8·a_5_21
  43. b_3_10·b_5_24 + b_4_15·b_4_17 + b_3_10·a_5_21 + b_3_8·a_5_21
  44. b_3_8·b_5_24 + b_3_8·a_5_21 + b_2_52·a_1_1·b_3_10 + b_2_52·a_1_1·b_3_8
  45. b_3_9·b_5_24 + b_4_15·b_4_17 + b_3_8·a_5_21 + b_2_52·a_1_1·b_3_10 + b_2_52·a_1_1·b_3_8
  46. b_3_10·a_5_21 + b_4_17·a_1_1·b_3_10 + b_2_5·a_1_1·b_5_25
  47. b_3_9·b_5_25 + b_3_8·b_5_25 + b_3_8·a_5_21 + b_2_52·a_1_1·b_3_10
  48. b_2_5·b_3_8·b_3_10 + b_2_5·b_6_37 + b_2_54 + b_3_10·a_5_21 + b_4_17·a_1_1·b_3_10
  49. b_3_10·a_5_21 + b_3_8·a_5_21 + b_4_17·a_1_1·b_3_10 + b_2_52·a_1_1·b_3_10
       + b_6_38·a_1_12
  50. b_4_15·b_4_17 + b_2_4·b_6_38 + b_2_42·b_4_15
  51. b_3_8·b_5_25 + b_2_52·b_4_17 + b_3_10·a_5_21 + b_3_8·a_5_21 + a_1_1·b_7_52
  52. a_1_0·b_7_52
  53. b_3_10·a_5_21 + a_1_1·b_7_53 + b_4_17·a_1_1·b_3_10 + b_2_52·a_1_1·b_3_8
  54. a_1_0·b_7_53
  55. b_4_15·a_5_21
  56. b_4_15·b_5_24 + b_2_4·b_4_17·b_3_9
  57. a_1_1·b_3_10·b_5_25 + b_4_17·a_5_21 + b_4_172·a_1_1
  58. b_4_15·b_5_25
  59. b_6_37·b_3_8 + b_2_53·b_3_10 + b_2_53·b_3_8 + b_2_52·a_5_21
  60. b_6_37·b_3_10 + b_6_37·b_3_9 + b_2_52·b_5_25 + b_2_53·b_3_8 + b_4_17·a_5_21
       + b_4_172·a_1_1 + b_2_5·b_6_38·a_1_1 + b_2_52·a_5_21 + b_2_54·a_1_1
  61. b_6_38·b_3_9 + b_6_38·b_3_8 + b_6_37·b_3_10 + b_6_37·b_3_9 + b_4_17·b_5_24
       + b_2_52·b_5_25 + b_2_53·b_3_8 + b_2_42·b_5_24 + b_4_17·a_5_21 + b_2_52·a_5_21
       + b_2_52·b_4_17·a_1_1 + b_2_54·a_1_1 + b_2_42·a_5_21
  62. b_6_38·b_3_8 + b_6_37·b_3_10 + b_6_37·b_3_9 + b_2_5·b_7_52 + b_2_5·b_4_17·b_3_10
       + b_2_52·b_5_25 + b_2_53·b_3_8 + b_4_17·a_5_21 + b_2_52·b_4_17·a_1_1
  63. a_1_12·b_7_52
  64. b_2_4·b_7_52 + b_2_4·b_4_17·b_3_9 + b_2_43·b_3_9
  65. b_6_37·b_3_10 + b_6_37·b_3_9 + b_2_5·b_7_53 + b_2_52·a_5_21 + b_2_54·a_1_1
  66. b_6_37·b_3_9 + b_2_53·b_3_10 + b_2_53·b_3_8 + b_2_4·b_7_53 + b_2_42·b_5_24
       + b_2_52·b_4_17·a_1_1 + b_2_54·a_1_1 + b_2_42·a_5_21
  67. b_8_68·a_1_1 + b_4_17·a_5_21 + b_4_172·a_1_1 + b_2_52·a_5_21 + b_2_52·b_4_17·a_1_1
       + b_2_54·a_1_1
  68. b_8_68·a_1_0
  69. a_5_212 + b_4_172·a_1_12
  70. b_5_242 + b_2_4·b_4_172 + b_4_172·a_1_12
  71. a_5_21·b_5_24 + b_4_172·a_1_12
  72. b_5_24·b_5_25 + b_4_17·a_1_1·b_5_25 + b_2_52·a_1_1·b_5_25 + b_4_172·a_1_12
  73. b_5_24·b_5_25 + a_5_21·b_5_25 + b_2_5·b_4_17·a_1_1·b_3_10 + b_2_52·a_1_1·b_5_25
       + b_4_172·a_1_12
  74. b_4_15·b_6_38 + b_2_4·b_4_172 + b_2_43·b_4_17
  75. b_5_24·b_5_25 + b_3_8·b_7_52 + b_2_5·b_3_10·b_5_25 + b_2_52·b_6_38
       + b_6_38·a_1_1·b_3_10 + b_2_5·b_4_17·a_1_1·b_3_10 + b_2_53·a_1_1·b_3_10
       + b_2_53·a_1_1·b_3_8
  76. b_5_24·b_5_25 + b_3_9·b_7_52 + b_2_5·b_3_10·b_5_25 + b_2_52·b_6_38 + b_2_4·b_4_172
       + b_2_43·b_4_17 + b_6_38·a_1_1·b_3_10 + b_2_5·a_1_1·b_7_52 + b_2_5·b_4_17·a_1_1·b_3_10
       + b_2_53·a_1_1·b_3_10 + b_2_53·a_1_1·b_3_8
  77. b_3_10·b_7_53 + b_4_17·b_6_37 + b_2_52·b_6_37 + b_2_53·b_4_17 + b_2_55
       + b_2_42·b_6_38 + b_2_43·b_4_15 + b_2_5·b_4_17·a_1_1·b_3_10
  78. b_3_8·b_7_53 + b_2_53·b_4_17 + b_2_55 + b_2_5·b_4_17·a_1_1·b_3_10
       + b_2_53·a_1_1·b_3_10 + b_2_53·a_1_1·b_3_8
  79. b_5_24·b_5_25 + b_3_9·b_7_53 + b_4_17·b_6_37 + b_2_5·b_3_10·b_5_25 + b_2_55
       + b_2_42·b_6_38 + b_2_43·b_4_15 + b_6_38·a_1_1·b_3_10 + b_2_52·a_1_1·b_5_25
       + b_2_53·a_1_1·b_3_10 + b_4_172·a_1_12
  80. b_5_252 + b_5_24·b_5_25 + b_2_5·b_4_172 + b_2_52·a_1_1·b_5_25 + b_4_172·a_1_12
       + c_8_72·a_1_12
  81. b_2_5·b_3_10·b_5_25 + b_2_5·b_8_68 + b_2_52·b_6_37 + b_2_5·a_1_1·b_7_52
       + b_2_52·a_1_1·b_5_25 + b_2_53·a_1_1·b_3_8
  82. b_4_15·b_6_37 + b_2_4·b_8_68 + b_2_43·b_4_15
  83. b_5_24·b_5_25 + a_1_1·b_9_95 + b_2_52·a_1_1·b_5_25 + b_2_53·a_1_1·b_3_10
       + b_4_172·a_1_12
  84. a_1_0·b_9_95
  85. b_6_38·b_5_24 + b_4_172·b_3_9 + b_2_5·b_4_17·b_5_25 + b_2_42·b_4_17·b_3_9
       + b_2_5·b_4_17·a_5_21 + b_2_5·b_4_172·a_1_1 + b_2_52·b_6_38·a_1_1
  86. b_6_37·b_5_25 + b_2_52·b_4_17·b_3_10 + b_2_53·b_5_25 + b_6_38·a_5_21
       + b_4_17·b_6_38·a_1_1 + b_2_5·b_4_172·a_1_1 + b_2_53·b_4_17·a_1_1
  87. b_6_37·b_5_25 + b_2_52·b_4_17·b_3_10 + b_2_53·b_5_25 + a_1_1·b_3_10·b_7_52
       + b_2_53·b_4_17·a_1_1
  88. b_4_15·b_7_52 + b_2_4·b_4_17·b_5_24 + b_2_43·b_5_24 + b_2_43·a_5_21
  89. b_6_37·b_5_24 + b_4_15·b_7_53 + b_2_42·b_4_17·b_3_9 + b_6_37·a_5_21
       + b_2_53·b_4_17·a_1_1 + b_2_55·a_1_1
  90. b_6_38·b_5_25 + b_6_38·b_5_24 + b_6_37·b_5_25 + b_4_17·b_7_52 + b_4_172·b_3_10
       + b_4_172·b_3_9 + b_2_5·b_4_17·b_5_25 + b_2_52·b_4_17·b_3_10 + b_2_53·b_5_25
       + b_4_17·b_6_38·a_1_1 + b_2_53·a_5_21 + b_2_55·a_1_1 + b_2_5·c_8_72·a_1_1
  91. b_6_37·a_5_21 + b_2_5·b_4_17·a_5_21 + b_2_5·b_4_172·a_1_1 + b_2_53·a_5_21
       + b_2_53·b_4_17·a_1_1 + b_2_4·c_8_72·a_1_0
  92. b_8_68·b_3_10 + b_6_38·b_5_24 + b_6_37·b_5_25 + b_6_37·b_5_24 + b_4_172·b_3_9
       + b_2_52·b_4_17·b_3_10 + b_2_54·b_3_10 + b_2_42·b_4_17·b_3_9 + b_2_43·b_5_24
       + b_6_37·a_5_21 + b_2_5·b_4_17·a_5_21 + b_2_55·a_1_1 + b_2_43·a_5_21
  93. b_8_68·b_3_8 + b_6_38·b_5_24 + b_6_37·b_5_25 + b_4_172·b_3_9 + b_2_5·b_4_17·b_5_25
       + b_2_53·b_5_25 + b_2_54·b_3_10 + b_2_54·b_3_8 + b_2_42·b_4_17·b_3_9
       + b_2_5·b_4_172·a_1_1 + b_2_53·a_5_21 + b_2_53·b_4_17·a_1_1 + b_2_55·a_1_1
  94. b_8_68·b_3_9 + b_6_38·b_5_24 + b_6_37·b_5_25 + b_6_37·b_5_24 + b_4_172·b_3_9
       + b_2_5·b_4_17·b_5_25 + b_2_53·b_5_25 + b_2_54·b_3_10 + b_2_54·b_3_8
       + b_2_42·b_4_17·b_3_9 + b_2_43·b_5_24 + b_6_37·a_5_21 + b_2_5·b_4_17·a_5_21
       + b_2_53·b_4_17·a_1_1 + b_2_55·a_1_1 + b_2_43·a_5_21
  95. b_6_38·b_5_24 + b_6_37·b_5_25 + b_4_172·b_3_9 + b_2_5·b_9_95 + b_2_52·b_4_17·b_3_10
       + b_2_53·b_5_25 + b_2_54·b_3_10 + b_2_42·b_4_17·b_3_9 + b_2_53·a_5_21 + b_2_55·a_1_1
  96. b_6_37·b_5_24 + b_2_4·b_9_95 + b_2_42·b_4_17·b_3_9 + b_2_43·b_5_24 + b_6_37·a_5_21
       + b_2_53·b_4_17·a_1_1 + b_2_55·a_1_1
  97. b_5_24·b_7_52 + b_2_4·b_4_17·b_6_38 + b_4_17·a_1_1·b_7_52 + b_2_52·a_1_1·b_7_52
  98. b_5_24·b_7_52 + b_2_4·b_4_17·b_6_38 + a_5_21·b_7_52 + b_2_5·b_6_38·a_1_1·b_3_10
       + b_2_5·b_4_17·a_1_1·b_5_25 + b_2_52·a_1_1·b_7_52
  99. a_5_21·b_7_53 + b_2_5·b_4_17·a_1_1·b_5_25 + b_2_52·b_4_17·a_1_1·b_3_10
       + b_2_53·a_1_1·b_5_25 + b_2_54·a_1_1·b_3_10
  100. b_6_382 + b_4_173 + b_2_5·b_4_17·b_6_38 + b_2_52·b_4_172 + b_2_53·b_6_38
       + b_2_53·b_6_37 + b_2_54·b_4_17 + b_2_44·b_4_17 + b_4_172·a_1_1·b_3_10
       + b_2_5·b_6_38·a_1_1·b_3_10 + b_2_5·b_4_17·a_1_1·b_5_25 + b_2_52·a_1_1·b_7_52
       + b_2_52·b_4_17·a_1_1·b_3_10 + b_2_54·a_1_1·b_3_8 + b_2_52·c_8_72
  101. b_5_25·b_7_53 + b_5_24·b_7_52 + b_2_52·b_4_172 + b_2_54·b_4_17 + b_2_4·b_4_17·b_6_38
       + b_2_5·b_4_17·a_1_1·b_5_25 + b_2_52·b_4_17·a_1_1·b_3_10 + b_2_54·a_1_1·b_3_10
       + b_4_17·b_6_38·a_1_12 + b_2_5·c_8_72·a_1_12
  102. b_5_25·b_7_53 + b_5_24·b_7_53 + b_6_37·b_6_38 + b_6_372 + b_2_5·b_3_10·b_7_52
       + b_2_53·b_6_38 + b_2_56 + b_2_42·b_4_172 + b_2_44·b_4_17
       + b_2_5·b_6_38·a_1_1·b_3_10 + b_2_5·b_4_17·a_1_1·b_5_25 + b_2_52·a_1_1·b_7_52
       + b_2_52·b_4_17·a_1_1·b_3_10 + b_2_53·a_1_1·b_5_25 + b_2_54·a_1_1·b_3_8
       + b_2_42·c_8_72
  103. b_5_25·b_7_53 + b_5_25·b_7_52 + b_5_24·b_7_52 + b_4_17·b_3_10·b_5_25
       + b_2_5·b_4_17·b_6_38 + b_2_52·b_4_172 + b_2_54·b_4_17 + b_2_4·b_4_17·b_6_38
       + b_2_5·b_6_38·a_1_1·b_3_10 + b_2_52·a_1_1·b_7_52 + b_2_52·b_4_17·a_1_1·b_3_10
       + b_2_54·a_1_1·b_3_8 + b_4_17·b_6_38·a_1_12 + c_8_72·a_1_1·b_3_8
  104. b_5_25·b_7_53 + b_5_24·b_7_53 + b_6_37·b_6_38 + b_2_5·b_3_10·b_7_52 + b_2_53·b_6_38
       + b_2_54·b_4_17 + b_2_42·b_8_68 + b_2_42·b_4_172 + b_2_44·b_4_15
       + b_2_5·b_6_38·a_1_1·b_3_10 + b_2_5·b_4_17·a_1_1·b_5_25 + b_2_52·a_1_1·b_7_52
       + b_2_52·b_4_17·a_1_1·b_3_10 + b_2_53·a_1_1·b_5_25 + b_2_54·a_1_1·b_3_8
  105. b_5_24·b_7_53 + b_5_24·b_7_52 + b_4_17·b_3_10·b_5_25 + b_4_17·b_8_68 + b_2_52·b_8_68
       + b_2_53·b_6_37 + b_2_54·b_4_17 + b_2_4·b_4_17·b_6_38 + b_2_42·b_4_172
       + b_2_43·b_6_38 + b_2_44·b_4_15 + b_2_5·b_6_38·a_1_1·b_3_10
       + b_2_5·b_4_17·a_1_1·b_5_25 + b_2_52·b_4_17·a_1_1·b_3_10 + b_2_53·a_1_1·b_5_25
       + b_4_17·b_6_38·a_1_12
  106. b_4_15·b_8_68 + b_2_4·b_4_17·b_6_37 + b_2_44·b_4_17
  107. b_5_24·b_7_53 + b_5_24·b_7_52 + b_3_10·b_9_95 + b_4_17·b_3_10·b_5_25 + b_2_54·b_4_17
       + b_2_4·b_4_17·b_6_38 + b_2_43·b_6_38 + b_2_44·b_4_15 + b_2_5·b_6_38·a_1_1·b_3_10
       + b_2_52·a_1_1·b_7_52 + b_2_54·a_1_1·b_3_10 + b_2_54·a_1_1·b_3_8
       + b_4_17·b_6_38·a_1_12
  108. b_5_24·b_7_52 + b_3_8·b_9_95 + b_2_52·b_4_172 + b_2_53·b_6_37 + b_2_56
       + b_2_4·b_4_17·b_6_38 + b_4_172·a_1_1·b_3_10 + b_2_5·b_6_38·a_1_1·b_3_10
       + b_2_5·b_4_17·a_1_1·b_5_25 + b_2_52·b_4_17·a_1_1·b_3_10 + b_2_53·a_1_1·b_5_25
       + b_2_54·a_1_1·b_3_10 + b_2_54·a_1_1·b_3_8 + b_4_17·b_6_38·a_1_12
  109. b_5_24·b_7_53 + b_5_24·b_7_52 + b_3_9·b_9_95 + b_2_52·b_4_172 + b_2_53·b_6_37
       + b_2_56 + b_2_4·b_4_17·b_6_38 + b_2_43·b_6_38 + b_2_44·b_4_15
       + b_4_172·a_1_1·b_3_10 + b_2_5·b_6_38·a_1_1·b_3_10 + b_2_5·b_4_17·a_1_1·b_5_25
       + b_2_52·b_4_17·a_1_1·b_3_10 + b_2_53·a_1_1·b_5_25
  110. b_6_37·b_7_52 + b_2_52·b_6_38·b_3_10 + b_2_52·b_4_17·b_5_25 + b_2_53·b_7_52
       + b_2_4·b_4_17·b_7_53 + b_2_42·b_4_17·b_5_24 + b_2_43·b_7_53 + b_2_44·b_5_24
       + b_2_52·b_4_17·a_5_21 + b_2_52·b_4_172·a_1_1 + b_2_54·a_5_21 + b_2_44·a_5_21
  111. b_6_38·b_7_52 + b_4_17·b_6_38·b_3_10 + b_4_172·b_5_25 + b_2_5·b_4_17·b_7_52
       + b_2_5·b_4_172·b_3_10 + b_2_52·b_4_17·b_5_25 + b_2_53·b_7_52
       + b_2_53·b_4_17·b_3_10 + b_2_54·b_5_25 + b_2_55·b_3_10 + b_2_55·b_3_8
       + b_2_42·b_4_17·b_5_24 + b_2_44·b_5_24 + b_4_173·a_1_1 + b_2_5·b_4_17·b_6_38·a_1_1
       + b_2_53·b_6_38·a_1_1 + b_2_56·a_1_1 + b_2_44·a_5_21 + b_2_5·c_8_72·b_3_8
       + b_2_52·c_8_72·a_1_1
  112. b_6_37·b_7_53 + b_2_53·b_4_17·b_3_10 + b_2_54·b_5_25 + b_2_55·b_3_10 + b_2_55·b_3_8
       + b_2_43·b_4_17·b_3_9 + b_2_52·b_4_17·a_5_21 + b_2_52·b_4_172·a_1_1
       + b_2_53·b_6_38·a_1_1 + b_2_54·a_5_21 + b_2_54·b_4_17·a_1_1 + b_2_4·c_8_72·b_3_9
  113. b_8_68·b_5_24 + b_6_37·b_7_52 + b_2_52·b_6_38·b_3_10 + b_2_52·b_4_17·b_5_25
       + b_2_53·b_7_52 + b_2_43·b_7_53 + b_2_43·b_4_17·b_3_9 + b_2_44·b_5_24
       + b_4_172·a_5_21 + b_4_173·a_1_1 + b_2_52·b_4_17·a_5_21 + b_2_52·b_4_172·a_1_1
       + b_2_56·a_1_1 + b_2_44·a_5_21
  114. b_8_68·b_5_25 + b_2_5·b_4_172·b_3_10 + b_2_53·b_4_17·b_3_10 + b_2_54·b_5_25
       + b_2_5·b_6_38·a_5_21
  115. b_8_68·a_5_21 + b_4_172·a_5_21 + b_4_173·a_1_1 + b_2_52·b_4_17·a_5_21
       + b_2_54·a_5_21 + b_2_54·b_4_17·a_1_1
  116. b_6_38·b_7_53 + b_4_17·b_9_95 + b_4_172·b_5_25 + b_2_5·b_4_17·b_7_52
       + b_2_5·b_4_172·b_3_10 + b_2_53·b_7_52 + b_2_42·b_9_95 + b_2_42·b_4_17·b_5_24
       + b_2_44·b_5_24 + b_4_172·a_5_21 + b_4_173·a_1_1 + b_2_5·b_6_38·a_5_21
       + b_2_52·b_4_172·a_1_1 + b_2_53·b_6_38·a_1_1 + b_2_54·a_5_21 + b_2_54·b_4_17·a_1_1
       + b_2_56·a_1_1
  117. b_6_37·b_7_52 + b_4_15·b_9_95 + b_2_52·b_6_38·b_3_10 + b_2_52·b_4_17·b_5_25
       + b_2_53·b_7_52 + b_2_42·b_4_17·b_5_24 + b_2_43·b_7_53 + b_2_43·b_4_17·b_3_9
       + b_2_44·b_5_24 + b_2_52·b_4_17·a_5_21 + b_2_52·b_4_172·a_1_1 + b_2_54·a_5_21
       + b_2_44·a_5_21
  118. b_7_522 + b_2_52·b_4_17·b_6_38 + b_2_53·b_4_172 + b_2_54·b_6_38 + b_2_54·b_6_37
       + b_2_55·b_4_17 + b_2_4·b_4_173 + b_2_45·b_4_17 + b_2_5·b_4_17·a_1_1·b_7_52
       + b_2_52·b_6_38·a_1_1·b_3_10 + b_2_54·a_1_1·b_5_25 + b_2_55·a_1_1·b_3_10
       + b_4_173·a_1_12 + b_2_53·c_8_72 + b_2_5·c_8_72·a_1_1·b_3_8
       + b_4_17·c_8_72·a_1_12
  119. b_7_52·b_7_53 + b_4_172·b_6_37 + b_2_52·b_4_17·b_6_38 + b_2_53·b_8_68
       + b_2_53·b_4_172 + b_2_54·b_6_38 + b_2_54·b_6_37 + b_2_42·b_4_17·b_6_38
       + b_2_42·b_4_17·b_6_37 + b_4_17·b_6_38·a_1_1·b_3_10 + b_4_172·a_1_1·b_5_25
       + b_2_5·b_4_17·a_1_1·b_7_52 + b_2_53·a_1_1·b_7_52 + b_2_54·a_1_1·b_5_25
       + b_2_55·a_1_1·b_3_10
  120. b_7_52·b_7_53 + b_6_37·b_8_68 + b_4_172·b_6_37 + b_2_52·b_4_17·b_6_38 + b_2_54·b_6_38
       + b_2_55·b_4_17 + b_2_57 + b_2_42·b_4_17·b_6_38 + b_2_43·b_8_68 + b_2_44·b_6_38
       + b_4_17·b_6_38·a_1_1·b_3_10 + b_4_172·a_1_1·b_5_25 + b_2_52·b_6_38·a_1_1·b_3_10
       + b_2_53·a_1_1·b_7_52 + b_2_54·a_1_1·b_5_25 + b_2_4·b_4_15·c_8_72
  121. b_7_532 + b_2_53·b_4_172 + b_2_57 + b_2_4·b_4_17·b_8_68 + b_2_44·b_6_38
       + b_2_45·b_4_15 + b_2_52·b_4_17·a_1_1·b_5_25 + b_2_55·a_1_1·b_3_8
       + b_2_4·b_4_17·c_8_72
  122. b_7_52·b_7_53 + b_7_522 + b_6_38·b_8_68 + b_4_17·b_3_10·b_7_52 + b_2_5·b_4_17·b_8_68
       + b_2_5·b_4_173 + b_2_52·b_3_10·b_7_52 + b_2_54·b_6_38 + b_2_54·b_6_37
       + b_2_42·b_4_17·b_6_38 + b_2_5·b_4_17·a_1_1·b_7_52 + b_2_5·b_4_172·a_1_1·b_3_10
       + b_2_53·b_4_17·a_1_1·b_3_10 + b_2_55·a_1_1·b_3_10 + b_4_173·a_1_12
       + b_2_53·c_8_72 + b_2_5·c_8_72·a_1_1·b_3_10 + b_4_17·c_8_72·a_1_12
  123. b_7_52·b_7_53 + b_5_24·b_9_95 + b_2_5·b_4_17·b_8_68 + b_2_52·b_4_17·b_6_38
       + b_2_54·b_6_38 + b_2_55·b_4_17 + b_2_42·b_4_17·b_6_37 + b_2_43·b_4_172
       + b_2_44·b_6_38 + b_2_45·b_4_15 + b_4_172·a_1_1·b_5_25 + b_2_5·b_4_172·a_1_1·b_3_10
       + b_2_52·b_6_38·a_1_1·b_3_10 + b_2_55·a_1_1·b_3_8 + b_4_173·a_1_12
  124. b_7_52·b_7_53 + b_5_25·b_9_95 + b_4_172·b_6_37 + b_2_5·b_4_173
       + b_2_52·b_4_17·b_6_38 + b_2_53·b_4_172 + b_2_54·b_6_38 + b_2_42·b_4_17·b_6_38
       + b_2_42·b_4_17·b_6_37 + b_4_172·a_1_1·b_5_25 + b_2_53·b_4_17·a_1_1·b_3_10
       + b_2_54·a_1_1·b_5_25 + b_2_55·a_1_1·b_3_10 + b_2_55·a_1_1·b_3_8
       + b_4_17·c_8_72·a_1_12
  125. a_5_21·b_9_95 + b_4_172·a_1_1·b_5_25 + b_2_5·b_4_172·a_1_1·b_3_10
       + b_2_53·b_4_17·a_1_1·b_3_10 + b_2_54·a_1_1·b_5_25
  126. b_8_68·b_7_53 + b_2_52·b_4_172·b_3_10 + b_2_55·b_5_25 + b_2_56·b_3_10
       + b_2_56·b_3_8 + b_2_43·b_9_95 + b_2_43·b_4_17·b_5_24 + b_2_45·b_5_24
       + b_4_17·b_6_38·a_5_21 + b_4_172·b_6_38·a_1_1 + b_2_5·b_4_173·a_1_1
       + b_2_52·b_6_38·a_5_21 + b_2_53·b_4_172·a_1_1 + b_2_55·a_5_21 + b_2_57·a_1_1
       + b_2_4·c_8_72·b_5_24 + b_2_4·c_8_72·a_5_21
  127. b_8_68·b_7_53 + b_6_37·b_9_95 + b_2_53·b_4_17·b_5_25 + b_2_56·b_3_8
       + b_2_44·b_4_17·b_3_9 + b_2_5·b_4_172·a_5_21 + b_2_5·b_4_173·a_1_1
       + b_2_52·b_6_38·a_5_21 + b_2_52·b_4_17·b_6_38·a_1_1 + b_2_43·c_8_72·a_1_0
  128. b_8_68·b_7_53 + b_8_68·b_7_52 + b_2_5·b_4_17·b_6_38·b_3_10 + b_2_5·b_4_172·b_5_25
       + b_2_52·b_4_172·b_3_10 + b_2_53·b_6_38·b_3_10 + b_2_53·b_4_17·b_5_25
       + b_2_54·b_7_52 + b_2_55·b_5_25 + b_2_56·b_3_10 + b_2_56·b_3_8 + b_2_4·b_4_17·b_9_95
       + b_2_42·b_4_172·b_3_9 + b_2_43·b_4_17·b_5_24 + b_2_44·b_4_17·b_3_9
       + b_2_45·b_5_24 + b_4_17·b_6_38·a_5_21 + b_4_172·b_6_38·a_1_1
       + b_2_53·b_4_172·a_1_1 + b_2_45·a_5_21 + b_2_4·c_8_72·b_5_24 + b_2_5·c_8_72·a_5_21
       + b_2_5·b_4_17·c_8_72·a_1_1 + b_2_53·c_8_72·a_1_1 + b_2_4·c_8_72·a_5_21
  129. b_6_38·b_9_95 + b_4_172·b_7_53 + b_4_172·b_7_52 + b_4_173·b_3_10
       + b_2_5·b_4_172·b_5_25 + b_2_53·b_6_38·b_3_10 + b_2_53·b_4_17·b_5_25
       + b_2_42·b_4_17·b_7_53 + b_2_44·b_4_17·b_3_9 + b_4_17·b_6_38·a_5_21
       + b_2_52·b_4_17·b_6_38·a_1_1 + b_2_53·b_4_17·a_5_21 + b_2_53·b_4_172·a_1_1
       + b_2_55·b_4_17·a_1_1 + b_2_57·a_1_1 + b_2_5·c_8_72·a_5_21 + b_2_53·c_8_72·a_1_1
  130. b_8_682 + b_2_52·b_4_173 + b_2_56·b_4_17 + b_2_58 + b_2_42·b_4_17·b_8_68
       + b_2_44·b_4_172 + b_2_45·b_6_38 + b_2_46·b_4_17 + b_2_46·b_4_15
       + b_2_42·b_4_17·c_8_72
  131. b_7_52·b_9_95 + b_8_682 + b_4_172·b_8_68 + b_2_5·b_4_172·b_6_38
       + b_2_52·b_4_17·b_8_68 + b_2_52·b_4_173 + b_2_53·b_3_10·b_7_52 + b_2_54·b_8_68
       + b_2_54·b_4_172 + b_2_55·b_6_37 + b_2_58 + b_2_42·b_4_173 + b_2_45·b_6_38
       + b_2_46·b_4_17 + b_2_46·b_4_15 + b_4_172·a_1_1·b_7_52
       + b_2_5·b_4_17·b_6_38·a_1_1·b_3_10 + b_2_5·b_4_172·a_1_1·b_5_25
       + b_2_52·b_4_17·a_1_1·b_7_52 + b_2_52·b_4_172·a_1_1·b_3_10
       + b_2_53·b_6_38·a_1_1·b_3_10 + b_2_54·a_1_1·b_7_52 + b_4_172·b_6_38·a_1_12
       + b_2_42·b_4_17·c_8_72 + b_2_5·c_8_72·a_1_1·b_5_25 + b_2_52·c_8_72·a_1_1·b_3_10
       + b_2_52·c_8_72·a_1_1·b_3_8 + b_6_38·c_8_72·a_1_12
  132. b_7_53·b_9_95 + b_8_682 + b_2_54·b_8_68 + b_2_54·b_4_172 + b_2_56·b_4_17
       + b_2_4·b_4_172·b_6_37 + b_2_46·b_4_17 + b_4_172·a_1_1·b_7_52
       + b_2_5·b_4_17·b_6_38·a_1_1·b_3_10 + b_2_52·b_4_172·a_1_1·b_3_10
       + b_2_55·a_1_1·b_5_25 + b_2_56·a_1_1·b_3_10 + b_4_172·b_6_38·a_1_12
       + b_2_4·b_6_38·c_8_72 + b_2_42·b_4_17·c_8_72 + b_2_42·b_4_15·c_8_72
  133. b_8_68·b_9_95 + b_2_5·b_4_173·b_3_10 + b_2_53·b_4_172·b_3_10 + b_2_56·b_5_25
       + b_2_57·b_3_10 + b_2_43·b_4_172·b_3_9 + b_2_44·b_4_17·b_5_24
       + b_2_45·b_4_17·b_3_9 + b_4_173·a_5_21 + b_4_174·a_1_1 + b_2_52·b_4_172·a_5_21
       + b_2_53·b_6_38·a_5_21 + b_2_54·b_4_172·a_1_1 + b_2_56·b_4_17·a_1_1 + b_2_58·a_1_1
       + b_2_4·b_4_17·c_8_72·b_3_9
  134. b_9_952 + b_2_5·b_4_174 + b_2_57·b_4_17 + b_2_4·b_4_172·b_8_68
       + b_2_44·b_4_17·b_6_38 + b_2_45·b_4_172 + b_2_46·b_6_38 + b_2_47·b_4_15
       + b_4_173·a_1_1·b_5_25 + b_2_56·a_1_1·b_5_25 + b_2_4·b_4_172·c_8_72
       + b_4_172·c_8_72·a_1_12

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 18.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_1_2, a Duflot regular element of degree 1
    2. c_8_72, a Duflot regular element of degree 8
    3. b_4_17 + b_2_42, an element of degree 4
    4. b_3_10 + b_3_8, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 9, 12].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. c_1_2 → c_1_0, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_3_9 → 0, an element of degree 3
  8. b_3_10 → 0, an element of degree 3
  9. b_4_15 → 0, an element of degree 4
  10. b_4_17 → 0, an element of degree 4
  11. a_5_21 → 0, an element of degree 5
  12. b_5_24 → 0, an element of degree 5
  13. b_5_25 → 0, an element of degree 5
  14. b_6_37 → 0, an element of degree 6
  15. b_6_38 → 0, an element of degree 6
  16. b_7_52 → 0, an element of degree 7
  17. b_7_53 → 0, an element of degree 7
  18. b_8_68 → 0, an element of degree 8
  19. c_8_72 → c_1_18, an element of degree 8
  20. b_9_95 → 0, an element of degree 9

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. c_1_2 → c_1_0, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_22, an element of degree 2
  6. b_3_8 → c_1_23, an element of degree 3
  7. b_3_9 → c_1_23, an element of degree 3
  8. b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_23, an element of degree 3
  9. b_4_15 → 0, an element of degree 4
  10. b_4_17 → c_1_34 + c_1_22·c_1_32 + c_1_24, an element of degree 4
  11. a_5_21 → 0, an element of degree 5
  12. b_5_24 → 0, an element of degree 5
  13. b_5_25 → c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_25, an element of degree 5
  14. b_6_37 → c_1_24·c_1_32 + c_1_25·c_1_3, an element of degree 6
  15. b_6_38 → c_1_36 + c_1_22·c_1_34 + c_1_24·c_1_32 + c_1_25·c_1_3 + c_1_26
       + c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
  16. b_7_52 → c_1_22·c_1_35 + c_1_23·c_1_34 + c_1_24·c_1_33 + c_1_25·c_1_32
       + c_1_12·c_1_25 + c_1_14·c_1_23, an element of degree 7
  17. b_7_53 → c_1_23·c_1_34 + c_1_25·c_1_32, an element of degree 7
  18. b_8_68 → c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_25·c_1_33 + c_1_26·c_1_32 + c_1_28, an element of degree 8
  19. c_8_72 → c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_24·c_1_34 + c_1_25·c_1_33
       + c_1_26·c_1_32 + c_1_27·c_1_3 + c_1_12·c_1_22·c_1_34
       + c_1_12·c_1_24·c_1_32 + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32
       + c_1_14·c_1_24 + c_1_18, an element of degree 8
  20. b_9_95 → c_1_2·c_1_38 + c_1_25·c_1_34 + c_1_27·c_1_32 + c_1_28·c_1_3, an element of degree 9

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. c_1_2 → c_1_0, an element of degree 1
  4. b_2_4 → c_1_22, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_3_9 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  8. b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  9. b_4_15 → c_1_22·c_1_32 + c_1_23·c_1_3, an element of degree 4
  10. b_4_17 → c_1_34 + c_1_22·c_1_32, an element of degree 4
  11. a_5_21 → 0, an element of degree 5
  12. b_5_24 → c_1_2·c_1_34 + c_1_23·c_1_32, an element of degree 5
  13. b_5_25 → 0, an element of degree 5
  14. b_6_37 → c_1_1·c_1_23·c_1_32 + c_1_1·c_1_24·c_1_3 + c_1_12·c_1_22·c_1_32
       + c_1_12·c_1_23·c_1_3 + c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
  15. b_6_38 → c_1_36 + c_1_2·c_1_35 + c_1_22·c_1_34 + c_1_23·c_1_33 + c_1_24·c_1_32
       + c_1_25·c_1_3, an element of degree 6
  16. b_7_52 → c_1_2·c_1_36 + c_1_22·c_1_35 + c_1_23·c_1_34 + c_1_24·c_1_33
       + c_1_25·c_1_32 + c_1_26·c_1_3, an element of degree 7
  17. b_7_53 → c_1_23·c_1_34 + c_1_25·c_1_32 + c_1_1·c_1_22·c_1_34 + c_1_1·c_1_24·c_1_32
       + c_1_12·c_1_2·c_1_34 + c_1_12·c_1_24·c_1_3 + c_1_14·c_1_2·c_1_32
       + c_1_14·c_1_22·c_1_3, an element of degree 7
  18. b_8_68 → c_1_26·c_1_32 + c_1_27·c_1_3 + c_1_1·c_1_23·c_1_34 + c_1_1·c_1_25·c_1_32
       + c_1_12·c_1_22·c_1_34 + c_1_12·c_1_25·c_1_3 + c_1_14·c_1_22·c_1_32
       + c_1_14·c_1_23·c_1_3, an element of degree 8
  19. c_8_72 → c_1_24·c_1_34 + c_1_26·c_1_32 + c_1_1·c_1_23·c_1_34 + c_1_1·c_1_25·c_1_32
       + c_1_12·c_1_24·c_1_32 + c_1_12·c_1_25·c_1_3 + c_1_14·c_1_34
       + c_1_14·c_1_23·c_1_3 + c_1_14·c_1_24 + c_1_18, an element of degree 8
  20. b_9_95 → c_1_23·c_1_36 + c_1_24·c_1_35 + c_1_26·c_1_33 + c_1_27·c_1_32
       + c_1_1·c_1_22·c_1_36 + c_1_1·c_1_23·c_1_35 + c_1_1·c_1_24·c_1_34
       + c_1_1·c_1_25·c_1_33 + c_1_12·c_1_2·c_1_36 + c_1_12·c_1_22·c_1_35
       + c_1_12·c_1_24·c_1_33 + c_1_12·c_1_25·c_1_32 + c_1_14·c_1_2·c_1_34
       + c_1_14·c_1_23·c_1_32, an element of degree 9

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128


Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009