Cohomology of group number 48 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 5.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 5.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 5 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t3  +  t2  −  t  +  1)
    (t  +  1)2 · (t  −  1)5 · (t2  +  1)
  • The a-invariants are -∞,-∞,-5,-5,-5,-5. 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 6:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. b_2_1, an element of degree 2
  4. b_2_2, an element of degree 2
  5. b_2_3, an element of degree 2
  6. c_2_4, a Duflot regular element of degree 2
  7. a_3_7, a nilpotent element of degree 3
  8. b_3_6, an element of degree 3
  9. b_3_8, an element of degree 3
  10. b_3_9, an element of degree 3
  11. b_4_10, an element of degree 4
  12. b_4_15, an element of degree 4
  13. b_4_16, an element of degree 4
  14. c_4_19, a Duflot regular element of degree 4
  15. b_5_29, an element of degree 5
  16. b_5_33, an element of degree 5
  17. b_6_39, an element of degree 6

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

Ring relations

There are 86 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_2_1·a_1_0
  4. b_2_2·a_1_0
  5. b_2_3·a_1_0
  6. b_2_22 + c_2_4·b_1_12
  7. a_1_0·a_3_7
  8. b_1_1·a_3_7
  9. a_1_0·b_3_6
  10. a_1_0·b_3_8
  11. a_1_0·b_3_9
  12. b_1_1·b_3_9 + b_2_1·b_2_2
  13. b_2_1·a_3_7
  14. b_2_2·a_3_7
  15. b_2_2·b_3_9 + b_2_1·c_2_4·b_1_1
  16. b_4_10·a_1_0
  17. b_4_10·b_1_1 + b_2_2·b_3_6
  18. b_4_15·a_1_0
  19. b_4_15·b_1_1 + b_2_2·b_3_8
  20. b_4_16·a_1_0
  21. b_4_16·b_1_1 + b_2_3·b_3_6 + b_2_32·b_1_1 + b_2_1·b_3_8
  22. a_3_72
  23. a_3_7·b_3_6
  24. b_3_62 + b_1_13·b_3_8 + b_2_1·b_1_1·b_3_6 + b_2_1·b_2_3·b_1_12 + b_2_13
  25. a_3_7·b_3_8
  26. a_3_7·b_3_9
  27. b_3_92 + b_2_12·c_2_4
  28. b_3_82 + b_2_3·b_1_1·b_3_8 + b_2_1·b_2_32 + c_4_19·b_1_12
  29. b_3_6·b_3_9 + b_2_1·b_4_10
  30. b_2_2·b_4_10 + c_2_4·b_1_1·b_3_6
  31. b_3_8·b_3_9 + b_2_1·b_4_15
  32. b_2_2·b_4_15 + c_2_4·b_1_1·b_3_8
  33. b_3_8·b_3_9 + b_2_3·b_4_10 + b_2_2·b_4_16 + b_2_2·b_2_32
  34. a_1_0·b_5_29
  35. b_3_6·b_3_8 + b_1_1·b_5_29 + b_2_3·b_1_1·b_3_8 + b_2_3·b_1_1·b_3_6 + b_2_12·b_2_3
  36. a_1_0·b_5_33
  37. b_3_8·b_3_9 + b_1_1·b_5_33 + b_2_3·b_4_10 + b_2_1·b_2_2·b_2_3
  38. b_4_10·a_3_7
  39. b_4_10·b_3_6 + b_2_2·b_1_12·b_3_8 + b_2_1·b_2_2·b_3_6 + b_2_1·b_2_2·b_2_3·b_1_1
       + b_2_12·b_3_9
  40. b_4_10·b_3_9 + b_2_1·c_2_4·b_3_6
  41. b_4_15·a_3_7
  42. b_4_15·b_3_6 + b_4_10·b_3_8
  43. b_4_15·b_3_8 + b_2_32·b_3_9 + b_2_2·b_2_3·b_3_8 + b_2_2·c_4_19·b_1_1
  44. b_4_15·b_3_9 + b_2_1·c_2_4·b_3_8
  45. b_4_16·a_3_7 + b_2_32·a_3_7
  46. b_4_16·b_3_6 + b_2_3·b_1_12·b_3_8 + b_2_32·b_3_6 + b_2_1·b_5_29 + b_2_1·b_2_3·b_3_8
       + b_2_1·b_2_32·b_1_1
  47. b_4_16·b_3_8 + b_2_3·b_5_29 + b_2_32·b_3_6 + b_2_1·b_2_3·b_3_8 + b_2_1·c_4_19·b_1_1
  48. b_4_10·b_3_8 + b_2_2·b_5_29 + b_2_2·b_2_3·b_3_8 + b_2_2·b_2_3·b_3_6 + b_2_1·b_2_3·b_3_9
  49. b_4_16·b_3_9 + b_2_32·b_3_9 + b_2_1·b_5_33 + b_2_1·b_2_3·b_3_9
  50. b_2_2·b_5_33 + b_2_3·c_2_4·b_3_6 + b_2_1·c_2_4·b_3_8 + b_2_1·b_2_3·c_2_4·b_1_1
  51. b_6_39·a_1_0
  52. b_6_39·b_1_1 + b_4_10·b_3_8 + b_2_1·b_2_3·b_3_9
  53. b_4_102 + c_2_4·b_1_13·b_3_8 + b_2_1·c_2_4·b_1_1·b_3_6 + b_2_1·b_2_3·c_2_4·b_1_12
       + b_2_13·c_2_4
  54. b_4_152 + b_2_3·c_2_4·b_1_1·b_3_8 + b_2_1·b_2_32·c_2_4 + c_2_4·c_4_19·b_1_12
  55. b_4_162 + b_2_32·b_1_1·b_3_8 + b_2_34 + b_2_1·b_2_3·b_4_16 + b_2_12·c_4_19
  56. a_3_7·b_5_29
  57. b_3_6·b_5_29 + b_2_3·b_1_1·b_5_29 + b_2_32·b_1_1·b_3_8 + b_2_32·b_1_1·b_3_6
       + b_2_1·b_1_1·b_5_29 + b_2_12·b_4_16 + c_4_19·b_1_14
  58. b_3_8·b_5_29 + b_2_32·b_1_1·b_3_8 + b_2_1·b_2_3·b_4_16 + c_4_19·b_1_1·b_3_6
       + b_2_3·c_4_19·b_1_12
  59. b_3_9·b_5_29 + b_4_10·b_4_16 + b_2_2·b_2_3·b_1_1·b_3_8 + b_2_2·b_2_3·b_4_16
       + b_2_2·b_2_33 + b_2_1·b_2_2·b_2_32
  60. b_4_10·b_4_15 + c_2_4·b_1_1·b_5_29 + b_2_3·c_2_4·b_1_1·b_3_8 + b_2_3·c_2_4·b_1_1·b_3_6
       + b_2_12·b_2_3·c_2_4
  61. a_3_7·b_5_33
  62. b_3_6·b_5_33 + b_4_10·b_4_16 + b_2_2·b_2_3·b_4_16 + b_2_2·b_2_33 + b_2_1·b_2_3·b_4_15
       + b_2_1·b_2_2·b_4_16 + b_2_1·b_2_2·b_2_32 + b_2_12·b_4_15
  63. b_3_8·b_5_33 + b_4_15·b_4_16 + b_2_32·b_4_15 + b_2_1·b_2_3·b_4_15
  64. b_3_9·b_5_33 + b_2_1·c_2_4·b_4_16 + b_2_1·b_2_32·c_2_4 + b_2_12·b_2_3·c_2_4
  65. b_4_10·b_4_16 + b_2_2·b_2_3·b_1_1·b_3_8 + b_2_2·b_2_3·b_4_16 + b_2_2·b_2_33
       + b_2_1·b_6_39 + b_2_1·b_2_3·b_4_15 + b_2_1·b_2_2·b_4_16 + b_2_12·b_4_15
  66. b_4_15·b_4_16 + b_2_3·b_6_39 + b_2_32·b_4_15 + b_2_1·b_2_3·b_4_15 + b_2_1·b_2_2·c_4_19
  67. b_4_10·b_4_15 + b_2_2·b_6_39 + b_2_12·b_2_3·c_2_4
  68. b_4_16·b_5_29 + b_2_33·b_3_8 + b_2_33·b_3_6 + b_2_1·b_2_3·b_5_29 + b_2_1·b_2_32·b_3_8
       + b_2_3·c_4_19·b_1_13 + b_2_1·c_4_19·b_3_6 + b_2_1·b_2_3·c_4_19·b_1_1
  69. b_4_15·b_5_33 + b_2_3·c_2_4·b_5_29 + b_2_32·c_2_4·b_3_8 + b_2_32·c_2_4·b_3_6
       + b_2_1·c_2_4·c_4_19·b_1_1
  70. b_4_10·b_5_29 + b_2_2·b_2_3·b_5_29 + b_2_2·b_2_32·b_3_8 + b_2_2·b_2_32·b_3_6
       + b_2_1·b_2_32·b_3_9 + b_2_1·b_2_2·b_5_29 + b_2_12·b_5_33 + b_2_12·b_2_3·b_3_9
       + b_2_2·c_4_19·b_1_13
  71. b_4_15·b_5_29 + b_2_33·b_3_9 + b_2_2·b_2_32·b_3_8 + b_2_1·b_2_3·b_5_33
       + b_2_1·b_2_32·b_3_9 + b_2_2·c_4_19·b_3_6 + b_2_2·b_2_3·c_4_19·b_1_1
  72. b_4_10·b_5_33 + b_2_3·c_2_4·b_1_12·b_3_8 + b_2_1·c_2_4·b_5_29
       + b_2_1·b_2_3·c_2_4·b_3_8 + b_2_1·b_2_3·c_2_4·b_3_6 + b_2_1·b_2_32·c_2_4·b_1_1
  73. b_4_16·b_5_33 + b_2_32·b_5_33 + b_2_33·b_3_9 + b_2_2·b_2_32·b_3_8
       + b_2_1·c_4_19·b_3_9
  74. b_6_39·a_3_7
  75. b_6_39·b_3_6 + b_4_10·b_5_29 + b_2_2·b_2_3·b_5_29 + b_2_2·b_2_3·b_1_12·b_3_8
       + b_2_2·b_2_32·b_3_8 + b_2_2·b_2_32·b_3_6 + b_2_1·b_2_32·b_3_9
       + b_2_1·b_2_2·b_2_3·b_3_6 + b_2_1·b_2_2·b_2_32·b_1_1 + b_2_12·b_2_3·b_3_9
  76. b_6_39·b_3_8 + b_4_15·b_5_29 + b_2_33·b_3_9 + b_2_2·b_2_3·b_5_29 + b_2_2·b_2_32·b_3_6
       + b_2_1·b_2_32·b_3_9 + b_2_2·b_2_3·c_4_19·b_1_1
  77. b_6_39·b_3_9 + b_2_1·c_2_4·b_5_29 + b_2_1·b_2_3·c_2_4·b_3_8 + b_2_1·b_2_3·c_2_4·b_3_6
  78. b_5_292 + b_2_33·b_1_1·b_3_8 + b_2_1·b_2_3·b_1_1·b_5_29 + b_2_1·b_2_32·b_1_1·b_3_8
       + b_2_1·b_2_34 + b_2_12·b_2_3·b_4_16 + c_4_19·b_1_13·b_3_8 + b_2_3·c_4_19·b_1_14
       + b_2_32·c_4_19·b_1_12 + b_2_1·c_4_19·b_1_1·b_3_6 + b_2_1·b_2_3·c_4_19·b_1_12
       + b_2_13·c_4_19
  79. b_5_332 + b_2_32·c_2_4·b_1_1·b_3_8 + b_2_1·b_2_3·c_2_4·b_4_16
       + b_2_12·b_2_32·c_2_4 + b_2_12·c_2_4·c_4_19
  80. b_4_15·b_6_39 + b_2_3·c_2_4·b_1_1·b_5_29 + b_2_32·c_2_4·b_1_1·b_3_8
       + b_2_32·c_2_4·b_1_1·b_3_6 + b_2_1·b_2_3·c_2_4·b_4_16 + b_2_1·b_2_33·c_2_4
       + b_2_12·b_2_32·c_2_4 + c_2_4·c_4_19·b_1_1·b_3_6
  81. b_5_29·b_5_33 + b_2_32·b_6_39 + b_2_1·b_2_32·b_4_15 + b_2_2·b_2_3·c_4_19·b_1_12
       + b_2_1·b_4_10·c_4_19 + b_2_1·b_2_2·b_2_3·c_4_19
  82. b_4_10·b_6_39 + b_2_3·c_2_4·b_1_13·b_3_8 + b_2_1·c_2_4·b_1_1·b_5_29
       + b_2_1·b_2_3·c_2_4·b_1_1·b_3_6 + b_2_1·b_2_32·c_2_4·b_1_12 + b_2_12·c_2_4·b_4_16
       + b_2_12·b_2_32·c_2_4 + b_2_13·b_2_3·c_2_4 + c_2_4·c_4_19·b_1_14
  83. b_5_29·b_5_33 + b_4_16·b_6_39 + b_2_2·b_2_32·b_1_1·b_3_8 + b_2_1·b_2_2·b_2_33
       + b_2_1·b_2_2·b_2_3·c_4_19
  84. b_6_39·b_5_33 + b_2_32·c_2_4·b_1_12·b_3_8 + b_2_1·b_2_3·c_2_4·b_5_29
       + b_2_1·b_2_32·c_2_4·b_3_6 + b_2_1·b_2_33·c_2_4·b_1_1 + b_2_3·c_2_4·c_4_19·b_1_13
       + b_2_1·c_2_4·c_4_19·b_3_6
  85. b_6_39·b_5_29 + b_2_2·b_2_32·b_5_29 + b_2_2·b_2_33·b_3_8 + b_2_2·b_2_33·b_3_6
       + b_2_1·b_2_32·b_5_33 + b_2_1·b_2_33·b_3_9 + b_2_1·b_2_2·b_2_32·b_3_8
       + b_2_2·c_4_19·b_1_12·b_3_8 + b_2_2·b_2_3·c_4_19·b_3_6 + b_2_1·b_2_2·c_4_19·b_3_6
       + b_2_1·b_2_2·b_2_3·c_4_19·b_1_1 + b_2_12·c_4_19·b_3_9
  86. b_6_392 + b_2_32·c_2_4·b_1_13·b_3_8 + b_2_1·b_2_3·c_2_4·b_1_1·b_5_29
       + b_2_1·b_2_32·c_2_4·b_1_1·b_3_8 + b_2_1·b_2_32·c_2_4·b_1_1·b_3_6
       + b_2_1·b_2_33·c_2_4·b_1_12 + b_2_12·b_2_3·c_2_4·b_4_16 + b_2_13·b_2_32·c_2_4
       + c_2_4·c_4_19·b_1_13·b_3_8 + b_2_3·c_2_4·c_4_19·b_1_14
       + b_2_1·c_2_4·c_4_19·b_1_1·b_3_6 + b_2_1·b_2_3·c_2_4·c_4_19·b_1_12
       + b_2_13·c_2_4·c_4_19

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 13.
  • However, the last relation was already found in degree 12 and the last generator in degree 6.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_4, a Duflot regular element of degree 2
    2. c_4_19, a Duflot regular element of degree 4
    3. b_1_1·b_3_8 + b_1_14 + b_2_32 + b_2_12, an element of degree 4
    4. b_1_13·b_3_8 + b_2_32·b_1_12 + b_2_1·b_1_1·b_3_8 + b_2_1·b_2_32 + b_2_12·b_1_12, an element of degree 6
    5. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 1, 5, 11, 13].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

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. b_1_1 → 0, an element of degree 1
  3. b_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. c_2_4 → c_1_02, an element of degree 2
  7. a_3_7 → 0, an element of degree 3
  8. b_3_6 → 0, an element of degree 3
  9. b_3_8 → 0, an element of degree 3
  10. b_3_9 → 0, an element of degree 3
  11. b_4_10 → 0, an element of degree 4
  12. b_4_15 → 0, an element of degree 4
  13. b_4_16 → 0, an element of degree 4
  14. c_4_19 → c_1_14, an element of degree 4
  15. b_5_29 → 0, an element of degree 5
  16. b_5_33 → 0, an element of degree 5
  17. b_6_39 → 0, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 5

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_2_1 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  4. b_2_2 → c_1_0·c_1_2, an element of degree 2
  5. b_2_3 → c_1_42 + c_1_3·c_1_4 + c_1_2·c_1_4 + c_1_1·c_1_2, an element of degree 2
  6. c_2_4 → c_1_02, an element of degree 2
  7. a_3_7 → 0, an element of degree 3
  8. b_3_6 → c_1_33 + c_1_2·c_1_3·c_1_4 + c_1_22·c_1_3 + c_1_1·c_1_22, an element of degree 3
  9. b_3_8 → c_1_3·c_1_42 + c_1_32·c_1_4 + c_1_2·c_1_3·c_1_4 + c_1_12·c_1_2, an element of degree 3
  10. b_3_9 → c_1_0·c_1_32 + c_1_0·c_1_2·c_1_3, an element of degree 3
  11. b_4_10 → c_1_0·c_1_33 + c_1_0·c_1_2·c_1_3·c_1_4 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_1·c_1_22, an element of degree 4
  12. b_4_15 → c_1_0·c_1_3·c_1_42 + c_1_0·c_1_32·c_1_4 + c_1_0·c_1_2·c_1_3·c_1_4
       + c_1_0·c_1_12·c_1_2, an element of degree 4
  13. b_4_16 → c_1_44 + c_1_3·c_1_43 + c_1_32·c_1_42 + c_1_33·c_1_4 + c_1_22·c_1_42
       + c_1_22·c_1_3·c_1_4 + c_1_1·c_1_33 + c_1_1·c_1_2·c_1_42 + c_1_1·c_1_22·c_1_4
       + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3, an element of degree 4
  14. c_4_19 → c_1_1·c_1_3·c_1_42 + c_1_1·c_1_32·c_1_4 + c_1_1·c_1_2·c_1_3·c_1_4
       + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_4
       + c_1_12·c_1_2·c_1_3 + c_1_13·c_1_2 + c_1_14, an element of degree 4
  15. b_5_29 → c_1_3·c_1_44 + c_1_32·c_1_43 + c_1_33·c_1_42 + c_1_34·c_1_4
       + c_1_2·c_1_3·c_1_43 + c_1_2·c_1_33·c_1_4 + c_1_22·c_1_3·c_1_42
       + c_1_22·c_1_32·c_1_4 + c_1_23·c_1_3·c_1_4 + c_1_1·c_1_34 + c_1_1·c_1_2·c_1_33
       + c_1_1·c_1_22·c_1_42 + c_1_1·c_1_22·c_1_32 + c_1_1·c_1_23·c_1_4
       + c_1_1·c_1_23·c_1_3 + c_1_12·c_1_33 + c_1_12·c_1_2·c_1_42
       + c_1_12·c_1_22·c_1_4 + c_1_12·c_1_22·c_1_3 + c_1_12·c_1_23, an element of degree 5
  16. b_5_33 → c_1_0·c_1_3·c_1_43 + c_1_0·c_1_32·c_1_42 + c_1_0·c_1_2·c_1_3·c_1_42
       + c_1_0·c_1_1·c_1_33 + c_1_0·c_1_1·c_1_2·c_1_42 + c_1_0·c_1_1·c_1_2·c_1_32
       + c_1_0·c_1_1·c_1_22·c_1_4 + c_1_0·c_1_12·c_1_32 + c_1_0·c_1_12·c_1_2·c_1_3
       + c_1_0·c_1_12·c_1_22, an element of degree 5
  17. b_6_39 → c_1_0·c_1_32·c_1_43 + c_1_0·c_1_33·c_1_42 + c_1_0·c_1_2·c_1_32·c_1_42
       + c_1_0·c_1_1·c_1_34 + c_1_0·c_1_1·c_1_2·c_1_3·c_1_42
       + c_1_0·c_1_1·c_1_2·c_1_32·c_1_4 + c_1_0·c_1_1·c_1_22·c_1_3·c_1_4
       + c_1_0·c_1_1·c_1_22·c_1_32 + c_1_0·c_1_12·c_1_33
       + c_1_0·c_1_12·c_1_2·c_1_3·c_1_4 + c_1_0·c_1_12·c_1_22·c_1_3
       + c_1_0·c_1_13·c_1_22, an element of degree 6

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