Cohomology of group number 135 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 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t6  +  t3  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-4,-3,-3. 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 17 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. b_2_2, an element of degree 2
  5. a_3_2, a nilpotent element of degree 3
  6. b_3_3, an element of degree 3
  7. b_3_4, an element of degree 3
  8. b_4_5, an element of degree 4
  9. b_4_6, an element of degree 4
  10. b_5_7, an element of degree 5
  11. b_5_8, an element of degree 5
  12. b_6_10, an element of degree 6
  13. b_6_11, an element of degree 6
  14. a_7_8, a nilpotent element of degree 7
  15. b_7_12, an element of degree 7
  16. b_8_17, an element of degree 8
  17. c_8_18, a Duflot regular element of degree 8

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

Ring relations

There are 105 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_1·a_1_0
  4. a_2_1·b_1_1
  5. b_2_2·a_1_0
  6. a_2_1·b_2_2
  7. a_1_0·a_3_2
  8. b_1_1·a_3_2
  9. a_1_0·b_3_3
  10. a_1_0·b_3_4 + a_2_12
  11. b_1_1·b_3_4
  12. b_2_2·a_3_2
  13. a_2_1·a_3_2
  14. a_2_1·b_3_3
  15. a_2_1·b_3_4
  16. b_4_5·a_1_0
  17. b_4_5·b_1_1
  18. b_4_6·b_1_1 + b_4_6·a_1_0
  19. a_3_22
  20. a_3_2·b_3_3
  21. b_3_32 + b_2_2·b_1_1·b_3_3 + b_2_23
  22. a_3_2·b_3_4
  23. b_3_3·b_3_4 + b_2_2·b_4_5
  24. a_2_1·b_4_5
  25. b_3_42 + b_2_2·b_4_6 + a_2_1·b_4_6
  26. a_1_0·b_5_7
  27. b_1_1·b_5_7
  28. a_1_0·b_5_8
  29. b_4_5·a_3_2
  30. b_4_5·b_3_3 + b_2_22·b_3_4
  31. b_4_6·b_3_3 + b_4_5·b_3_4
  32. b_4_5·b_3_4 + b_2_2·b_5_7 + b_2_22·b_3_4
  33. a_2_1·b_5_7
  34. a_2_1·b_5_8
  35. b_6_10·a_1_0
  36. b_6_10·b_1_1 + b_2_2·b_5_8
  37. b_6_11·a_1_0 + b_4_6·a_3_2
  38. b_6_11·b_1_1 + b_2_2·b_5_8 + b_4_6·a_3_2
  39. b_4_52 + b_2_22·b_4_6
  40. a_3_2·b_5_7
  41. b_3_3·b_5_7 + b_4_52 + b_2_22·b_4_5
  42. a_3_2·b_5_8
  43. b_3_4·b_5_8
  44. a_2_1·b_6_10
  45. b_4_5·b_4_6 + b_2_2·b_6_11 + b_2_2·b_6_10
  46. b_3_4·b_5_7 + b_4_5·b_4_6 + b_4_52 + a_2_1·b_6_11
  47. a_1_0·a_7_8
  48. b_1_1·a_7_8
  49. a_1_0·b_7_12
  50. b_3_3·b_5_8 + b_1_1·b_7_12 + b_1_13·b_5_8 + b_2_2·b_1_1·b_5_8
  51. b_4_5·b_5_7 + b_2_2·b_4_6·b_3_4 + b_2_22·b_5_7 + b_2_23·b_3_4
  52. b_4_5·b_5_8
  53. b_6_10·a_3_2
  54. b_6_11·a_3_2 + b_4_62·a_1_0
  55. b_6_11·b_3_4 + b_6_10·b_3_4 + b_4_6·b_5_8 + b_4_6·b_5_7 + b_2_2·b_4_6·b_3_4
       + b_4_62·a_1_0
  56. b_6_11·b_3_3 + b_6_10·b_3_3 + b_2_2·b_4_6·b_3_4
  57. b_2_2·a_7_8
  58. a_2_1·a_7_8
  59. b_6_10·b_3_3 + b_2_2·b_7_12 + b_2_2·b_1_12·b_5_8 + b_2_22·b_5_8 + b_2_22·b_5_7
  60. a_2_1·b_7_12
  61. b_8_17·a_1_0 + b_4_62·a_1_0
  62. b_8_17·b_1_1 + b_2_2·b_1_12·b_5_8 + b_2_22·b_5_8 + b_4_62·a_1_0
  63. b_5_72 + b_2_2·b_4_62 + b_2_23·b_4_6 + a_2_1·b_4_62
  64. b_5_7·b_5_8
  65. b_4_5·b_6_11 + b_4_5·b_6_10 + b_2_2·b_4_62
  66. a_3_2·a_7_8
  67. b_3_4·a_7_8
  68. b_3_3·a_7_8
  69. a_3_2·b_7_12
  70. b_3_4·b_7_12 + b_4_5·b_6_10 + b_2_22·b_6_11 + b_2_22·b_6_10 + b_2_23·b_4_6
  71. b_3_3·b_7_12 + b_1_13·b_7_12 + b_1_15·b_5_8 + b_2_2·b_1_13·b_5_8 + b_2_22·b_6_10
       + b_2_23·b_4_6 + b_2_23·b_4_5
  72. b_5_82 + b_2_22·b_1_1·b_5_8 + c_8_18·b_1_12
  73. b_4_5·b_6_10 + b_2_2·b_8_17 + b_2_2·b_4_62 + b_2_22·b_1_1·b_5_8 + b_2_22·b_6_11
       + b_2_23·b_4_6
  74. a_2_1·b_8_17 + a_2_1·b_4_62
  75. b_6_11·b_5_8 + b_6_11·b_5_7 + b_6_10·b_5_8 + b_6_10·b_5_7 + b_4_62·b_3_4
       + b_2_2·b_4_6·b_5_7 + b_2_22·b_4_6·b_3_4 + b_4_62·a_3_2
  76. b_6_11·b_5_8 + b_6_10·b_5_8 + b_4_6·a_7_8
  77. b_4_5·a_7_8
  78. b_6_10·b_5_7 + b_4_6·b_7_12 + b_2_2·b_6_10·b_3_4 + b_2_2·b_4_6·b_5_7 + b_4_62·a_3_2
  79. b_4_5·b_7_12 + b_2_2·b_6_10·b_3_4 + b_2_22·b_4_6·b_3_4 + b_2_23·b_5_7 + b_2_24·b_3_4
  80. b_6_10·b_5_8 + b_2_23·b_5_8 + b_2_2·c_8_18·b_1_1
  81. b_8_17·a_3_2 + b_4_62·a_3_2
  82. b_8_17·b_3_4 + b_6_10·b_5_7 + b_4_62·b_3_4 + b_2_2·b_4_6·b_5_7
  83. b_8_17·b_3_3 + b_2_2·b_1_12·b_7_12 + b_2_2·b_1_14·b_5_8 + b_2_2·b_6_10·b_3_4
       + b_2_2·b_4_6·b_5_7 + b_2_22·b_7_12 + b_2_23·b_5_8 + b_2_24·b_3_4
  84. b_5_8·a_7_8
  85. b_5_7·a_7_8
  86. b_5_7·b_7_12 + b_6_102 + b_2_2·b_4_6·b_6_10 + b_2_23·b_6_11 + b_2_22·c_8_18
  87. b_6_112 + b_6_102 + b_4_63 + a_2_1·b_4_6·b_6_11 + a_2_12·c_8_18
  88. b_5_8·b_7_12 + b_2_22·b_1_1·b_7_12 + c_8_18·b_1_1·b_3_3 + c_8_18·b_1_14
       + b_2_2·c_8_18·b_1_12
  89. b_5_7·b_7_12 + b_2_2·b_4_6·b_6_10 + b_2_22·b_8_17 + b_2_23·b_1_1·b_5_8
       + b_2_23·b_6_11
  90. b_6_10·b_6_11 + b_6_102 + b_4_6·b_8_17 + b_4_63 + b_2_2·b_4_6·b_6_11
       + b_2_22·b_4_62 + a_2_1·b_4_6·b_6_11
  91. b_5_7·b_7_12 + b_4_5·b_8_17 + b_2_2·b_4_6·b_6_11 + b_2_2·b_4_6·b_6_10 + b_2_23·b_6_11
       + b_2_23·b_6_10 + b_2_24·b_4_6
  92. b_4_62·b_5_8 + b_6_11·a_7_8
  93. b_6_10·a_7_8
  94. b_6_11·b_7_12 + b_6_10·b_7_12 + b_4_6·b_6_10·b_3_4 + b_2_2·b_4_62·b_3_4
       + b_2_22·b_4_6·b_5_7 + b_2_23·b_4_6·b_3_4 + b_4_63·a_1_0
  95. b_6_11·b_7_12 + b_4_6·b_6_10·b_3_4 + b_2_2·b_4_6·b_7_12 + b_2_2·b_4_62·b_3_4
       + b_2_22·b_4_6·b_5_7 + b_2_23·b_7_12 + b_2_23·b_4_6·b_3_4 + b_2_25·b_3_4
       + b_4_63·a_1_0 + b_2_2·c_8_18·b_3_3 + b_2_2·c_8_18·b_1_13 + b_2_22·c_8_18·b_1_1
  96. b_8_17·b_5_8 + b_4_62·b_5_8 + b_2_23·b_1_12·b_5_8 + b_2_24·b_5_8
       + b_2_2·c_8_18·b_1_13 + b_2_22·c_8_18·b_1_1
  97. b_8_17·b_5_7 + b_4_6·b_6_10·b_3_4 + b_4_62·b_5_7 + b_2_2·b_4_62·b_3_4
       + b_2_22·b_6_10·b_3_4 + b_2_23·b_4_6·b_3_4
  98. a_7_82
  99. a_7_8·b_7_12
  100. b_7_122 + b_2_22·b_1_15·b_5_8 + b_2_23·b_1_1·b_7_12 + b_2_23·b_1_13·b_5_8
       + b_2_23·b_8_17 + b_2_23·b_4_62 + b_2_24·b_1_1·b_5_8 + b_2_25·b_4_6
       + c_8_18·b_1_16 + b_2_2·c_8_18·b_1_1·b_3_3 + b_2_22·c_8_18·b_1_12 + b_2_23·c_8_18
  101. b_6_11·b_8_17 + b_4_62·b_6_11 + b_4_62·b_6_10 + b_2_2·b_4_63 + b_2_24·b_1_1·b_5_8
       + b_2_24·b_6_10 + b_2_25·b_4_6 + a_2_1·b_4_63 + b_2_2·b_4_5·c_8_18
       + b_2_22·c_8_18·b_1_12 + b_2_23·c_8_18
  102. b_6_10·b_8_17 + b_4_62·b_6_10 + b_2_2·b_4_6·b_8_17 + b_2_2·b_4_63
       + b_2_22·b_4_6·b_6_10 + b_2_23·b_4_62 + b_2_24·b_1_1·b_5_8 + b_2_24·b_6_10
       + b_2_25·b_4_6 + b_2_2·b_4_5·c_8_18 + b_2_22·c_8_18·b_1_12 + b_2_23·c_8_18
  103. b_8_17·b_7_12 + b_4_62·b_7_12 + b_2_23·b_1_12·b_7_12 + b_2_23·b_6_10·b_3_4
       + b_2_24·b_7_12 + b_2_24·b_4_6·b_3_4 + b_2_26·b_3_4 + b_2_2·c_8_18·b_1_12·b_3_3
       + b_2_2·c_8_18·b_1_15 + b_2_22·c_8_18·b_3_4 + b_2_22·c_8_18·b_3_3
       + b_2_23·c_8_18·b_1_1
  104. b_8_17·a_7_8 + b_4_62·a_7_8
  105. b_8_172 + b_4_64 + b_2_22·b_4_6·b_8_17 + b_2_22·b_4_63 + b_2_24·b_1_13·b_5_8
       + b_2_24·b_8_17 + b_2_24·b_4_62 + b_2_25·b_1_1·b_5_8 + b_2_22·c_8_18·b_1_14
       + b_2_22·b_4_6·c_8_18 + b_2_24·c_8_18

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 16.
  • 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_8_18, a Duflot regular element of degree 8
    2. b_1_14 + b_4_6 + b_2_22, an element of degree 4
    3. b_3_4 + b_2_2·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 9, 12].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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 1

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. b_3_3 → 0, an element of degree 3
  7. b_3_4 → 0, an element of degree 3
  8. b_4_5 → 0, an element of degree 4
  9. b_4_6 → 0, an element of degree 4
  10. b_5_7 → 0, an element of degree 5
  11. b_5_8 → 0, an element of degree 5
  12. b_6_10 → 0, an element of degree 6
  13. b_6_11 → 0, an element of degree 6
  14. a_7_8 → 0, an element of degree 7
  15. b_7_12 → 0, an element of degree 7
  16. b_8_17 → 0, an element of degree 8
  17. c_8_18 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. b_3_3 → c_1_23 + c_1_12·c_1_2, an element of degree 3
  7. b_3_4 → 0, an element of degree 3
  8. b_4_5 → 0, an element of degree 4
  9. b_4_6 → 0, an element of degree 4
  10. b_5_7 → 0, an element of degree 5
  11. b_5_8 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  12. b_6_10 → c_1_0·c_1_1·c_1_24 + c_1_0·c_1_13·c_1_22 + c_1_02·c_1_24
       + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_22 + c_1_04·c_1_1·c_1_2, an element of degree 6
  13. b_6_11 → c_1_0·c_1_1·c_1_24 + c_1_0·c_1_13·c_1_22 + c_1_02·c_1_24
       + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_22 + c_1_04·c_1_1·c_1_2, an element of degree 6
  14. a_7_8 → 0, an element of degree 7
  15. b_7_12 → c_1_0·c_1_1·c_1_25 + c_1_0·c_1_13·c_1_23 + c_1_0·c_1_14·c_1_22
       + c_1_0·c_1_15·c_1_2 + c_1_02·c_1_25 + c_1_02·c_1_14·c_1_2 + c_1_02·c_1_15
       + c_1_04·c_1_23 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_13, an element of degree 7
  16. b_8_17 → c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_14·c_1_23
       + c_1_0·c_1_15·c_1_22 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_22
       + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2, an element of degree 8
  17. c_8_18 → c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_13·c_1_24
       + c_1_0·c_1_14·c_1_23 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_13·c_1_23 + c_1_04·c_1_14 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_12, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. b_3_3 → c_1_13, an element of degree 3
  7. b_3_4 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_4_5 → c_1_12·c_1_22 + c_1_13·c_1_2, an element of degree 4
  9. b_4_6 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  10. b_5_7 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  11. b_5_8 → 0, an element of degree 5
  12. b_6_10 → c_1_12·c_1_24 + c_1_15·c_1_2 + c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14
       + c_1_04·c_1_12, an element of degree 6
  13. b_6_11 → c_1_26 + c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_15·c_1_2 + c_1_0·c_1_13·c_1_22
       + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2
       + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  14. a_7_8 → 0, an element of degree 7
  15. b_7_12 → c_1_0·c_1_14·c_1_22 + c_1_0·c_1_15·c_1_2 + c_1_02·c_1_13·c_1_22
       + c_1_02·c_1_14·c_1_2 + c_1_02·c_1_15 + c_1_04·c_1_13, an element of degree 7
  16. b_8_17 → c_1_28 + c_1_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14, an element of degree 8
  17. c_8_18 → c_1_16·c_1_22 + c_1_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + c_1_08, an element of degree 8

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