Cohomology of group number 515 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 5.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all 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
    ( − 2) · (t2  −  1/2·t  +  1/2)
    (t  +  1) · (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 5:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. b_2_3, an element of degree 2
  5. b_2_4, an element of degree 2
  6. b_2_5, an element of degree 2
  7. b_2_6, an element of degree 2
  8. c_2_7, a Duflot regular element of degree 2
  9. b_3_14, an element of degree 3
  10. b_3_15, an element of degree 3
  11. b_3_16, an element of degree 3
  12. b_3_17, an element of degree 3
  13. b_4_30, an element of degree 4
  14. b_4_31, an element of degree 4
  15. b_4_32, an element of degree 4
  16. c_4_35, a Duflot regular element of degree 4
  17. b_5_58, an element of degree 5

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 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_1_0·b_1_2
  4. b_2_3·a_1_0
  5. b_2_4·a_1_0
  6. b_2_4·b_1_1 + b_2_3·b_1_2
  7. b_2_5·a_1_0
  8. b_2_6·a_1_0
  9. b_2_3·b_1_12 + b_2_32
  10. b_2_3·b_1_1·b_1_2 + b_2_3·b_2_4
  11. b_2_42 + b_2_3·b_1_22
  12. b_2_6·b_1_22 + b_2_5·b_1_1·b_1_2 + b_2_52 + c_2_7·b_1_12
  13. b_1_2·b_3_14 + b_2_4·b_2_5
  14. a_1_0·b_3_14
  15. b_1_1·b_3_14 + b_2_3·b_2_5
  16. b_1_2·b_3_15 + b_2_4·b_2_6
  17. a_1_0·b_3_15
  18. b_1_1·b_3_15 + b_2_3·b_2_6
  19. a_1_0·b_3_16
  20. a_1_0·b_3_17
  21. b_1_2·b_3_16 + b_1_1·b_3_17 + b_2_5·b_2_6
  22. b_2_3·b_3_14 + b_2_3·b_2_5·b_1_1
  23. b_2_5·b_3_14 + b_2_4·b_2_6·b_1_2 + b_2_3·b_2_5·b_1_2 + b_2_3·c_2_7·b_1_1
  24. b_2_4·b_3_14 + b_2_3·b_2_5·b_1_2
  25. b_2_3·b_3_15 + b_2_3·b_2_6·b_1_1
  26. b_2_6·b_3_14 + b_2_5·b_3_15
  27. b_2_4·b_3_15 + b_2_3·b_2_6·b_1_2
  28. b_2_6·b_3_14 + b_2_4·b_3_16 + b_2_3·b_3_17
  29. b_4_30·b_1_2 + b_2_4·b_3_17
  30. b_4_30·a_1_0
  31. b_4_30·b_1_1 + b_2_6·b_3_14 + b_2_4·b_3_16
  32. b_4_31·b_1_2 + b_2_62·b_1_2 + b_2_4·b_3_16
  33. b_4_31·a_1_0
  34. b_4_31·b_1_1 + b_2_62·b_1_1 + b_2_3·b_3_16
  35. b_4_32·b_1_2 + b_2_5·b_3_17 + b_2_5·b_2_6·b_1_2 + b_2_4·b_3_17 + b_2_6·c_2_7·b_1_1
  36. b_4_32·a_1_0
  37. b_4_32·b_1_1 + b_2_6·b_3_14 + b_2_62·b_1_2 + b_2_5·b_3_16 + b_2_4·b_3_16
  38. b_3_142 + b_2_3·b_2_52
  39. b_3_14·b_3_15 + b_2_3·b_2_5·b_2_6
  40. b_3_152 + b_2_3·b_2_62
  41. b_3_172 + b_2_6·b_1_2·b_3_17 + b_2_4·b_2_5·b_2_6 + b_2_3·b_1_2·b_3_17
       + c_4_35·b_1_22 + b_2_62·c_2_7
  42. b_3_162 + b_2_6·b_1_1·b_3_16 + b_2_63 + b_2_3·b_1_1·b_3_16 + c_4_35·b_1_12
  43. b_2_3·b_1_1·b_3_17 + b_2_3·b_4_30
  44. b_3_14·b_3_17 + b_2_5·b_4_30
  45. b_2_4·b_4_30 + b_2_3·b_1_2·b_3_17
  46. b_3_15·b_3_17 + b_2_6·b_4_30
  47. b_2_3·b_1_1·b_3_16 + b_2_3·b_4_31 + b_2_3·b_2_62
  48. b_3_14·b_3_16 + b_2_5·b_4_31 + b_2_5·b_2_62
  49. b_2_4·b_4_31 + b_2_4·b_2_62 + b_2_3·b_1_1·b_3_17 + b_2_3·b_2_5·b_2_6
  50. b_3_15·b_3_16 + b_2_6·b_4_31 + b_2_63
  51. b_3_14·b_3_16 + b_2_4·b_2_62 + b_2_3·b_1_1·b_3_17 + b_2_3·b_4_32
  52. b_3_14·b_3_17 + b_2_6·b_1_2·b_3_17 + b_2_5·b_1_1·b_3_17 + b_2_5·b_4_32 + b_2_52·b_2_6
       + c_2_7·b_1_1·b_3_16
  53. b_3_14·b_3_17 + b_2_4·b_4_32 + b_2_4·b_2_5·b_2_6 + b_2_3·b_1_2·b_3_17
       + b_2_3·b_2_6·c_2_7
  54. b_3_16·b_3_17 + b_3_15·b_3_17 + b_2_6·b_1_1·b_3_17 + b_2_6·b_4_32 + b_2_5·b_2_62
       + b_2_3·b_1_1·b_3_17 + b_2_3·b_2_5·b_2_6 + c_4_35·b_1_1·b_1_2
  55. b_3_14·b_3_17 + b_1_2·b_5_58 + b_2_4·b_2_5·b_2_6 + b_2_3·b_2_6·c_2_7
  56. a_1_0·b_5_58
  57. b_3_14·b_3_16 + b_1_1·b_5_58 + b_2_4·b_2_62
  58. b_4_30·b_3_17 + b_2_4·b_2_6·b_3_17 + b_2_3·b_2_5·b_2_6·b_1_2 + b_2_3·b_2_4·b_3_17
       + b_2_6·c_2_7·b_3_15 + b_2_4·c_4_35·b_1_2
  59. b_4_30·b_3_14 + b_2_3·b_2_5·b_3_17
  60. b_4_30·b_3_15 + b_2_3·b_2_6·b_3_17
  61. b_4_31·b_3_17 + b_4_30·b_3_16 + b_2_62·b_3_17
  62. b_4_31·b_3_16 + b_2_62·b_3_16 + b_2_62·b_3_15 + b_2_3·b_2_6·b_3_16 + b_2_32·b_3_16
       + b_2_3·c_4_35·b_1_1
  63. b_4_31·b_3_14 + b_2_4·b_2_6·b_3_16 + b_2_3·b_2_6·b_3_17 + b_2_3·b_2_5·b_3_16
  64. b_4_31·b_3_15 + b_2_62·b_3_15 + b_2_3·b_2_6·b_3_16
  65. b_4_32·b_3_17 + b_2_4·b_2_6·b_3_17 + b_2_4·b_2_5·b_3_16 + b_2_3·b_2_5·b_2_6·b_1_2
       + b_2_3·b_2_4·b_3_17 + b_2_6·c_2_7·b_3_16 + b_2_6·c_2_7·b_3_15 + b_2_5·c_4_35·b_1_2
       + b_2_4·c_4_35·b_1_2
  66. b_4_32·b_3_16 + b_4_30·b_3_16 + b_2_62·b_3_17 + b_2_5·b_2_6·b_3_16 + b_2_3·b_2_5·b_3_16
       + b_2_5·c_4_35·b_1_1
  67. b_4_32·b_3_14 + b_2_4·b_2_6·b_3_17 + b_2_4·b_2_5·b_3_16 + b_2_3·b_2_5·b_3_17
       + b_2_3·c_2_7·b_3_16
  68. b_4_32·b_3_15 + b_4_30·b_3_16 + b_2_4·b_2_6·b_3_16 + b_2_3·b_2_6·b_3_17
       + b_2_3·b_2_4·b_3_16 + b_2_3·c_4_35·b_1_2
  69. b_2_3·b_5_58 + b_2_3·b_2_62·b_1_2 + b_2_3·b_2_5·b_3_16
  70. b_2_5·b_5_58 + b_2_4·b_2_6·b_3_17 + b_2_4·b_2_5·b_3_16 + b_2_3·c_2_7·b_3_16
  71. b_2_4·b_5_58 + b_2_3·b_2_5·b_3_17 + b_2_3·b_2_5·b_2_6·b_1_2 + b_2_3·b_2_6·c_2_7·b_1_1
  72. b_4_30·b_3_16 + b_2_6·b_5_58 + b_2_4·b_2_6·b_3_16 + b_2_3·b_2_4·b_3_16
       + b_2_3·c_4_35·b_1_2
  73. b_4_312 + b_2_64 + b_2_3·b_2_6·b_4_31 + b_2_32·b_4_31 + b_2_32·b_2_62
       + b_2_32·c_4_35
  74. b_4_302 + b_2_3·b_2_5·b_4_32 + b_2_3·b_2_52·b_2_6 + b_2_3·b_2_4·b_2_5·b_2_6
       + b_2_32·b_1_2·b_3_17 + b_2_3·c_4_35·b_1_22 + b_2_3·c_2_7·b_4_31
  75. b_4_30·b_4_31 + b_2_62·b_4_30 + b_2_3·b_2_6·b_4_32 + b_2_3·b_2_5·b_2_62
       + b_2_32·b_4_30 + b_2_32·b_2_5·b_2_6 + b_2_3·b_2_4·c_4_35
  76. b_4_322 + b_4_302 + b_2_5·b_2_6·b_4_32 + b_2_4·b_2_6·b_4_32 + b_2_3·b_2_52·b_2_6
       + b_2_3·b_2_4·b_4_32 + b_2_3·b_2_4·b_2_5·b_2_6 + b_2_32·b_1_2·b_3_17 + b_2_63·c_2_7
       + b_2_52·c_4_35 + b_2_3·c_2_7·b_4_31 + b_2_32·b_2_6·c_2_7
  77. b_4_31·b_4_32 + b_2_62·b_4_32 + b_2_62·b_4_30 + b_2_4·b_2_63 + b_2_3·b_2_6·b_4_30
       + b_2_3·b_2_5·b_2_62 + b_2_3·b_2_4·b_2_62 + b_2_32·b_4_32 + b_2_32·b_2_5·b_2_6
       + b_2_3·b_2_5·c_4_35 + b_2_3·b_2_4·c_4_35
  78. b_4_30·b_4_32 + b_4_302 + b_2_3·b_2_52·b_2_6 + b_2_3·b_2_4·b_4_32
       + b_2_3·b_2_4·b_2_5·b_2_6 + b_2_32·b_1_2·b_3_17 + b_2_6·c_2_7·b_4_31 + b_2_63·c_2_7
       + b_2_4·b_2_5·c_4_35 + b_2_32·b_2_6·c_2_7
  79. b_3_17·b_5_58 + b_2_3·b_2_52·b_2_6 + b_2_3·b_2_4·b_4_32 + b_2_3·b_2_4·b_2_5·b_2_6
       + b_2_32·b_1_2·b_3_17 + b_2_6·c_2_7·b_4_31 + b_2_63·c_2_7 + b_2_4·b_2_5·c_4_35
       + b_2_32·b_2_6·c_2_7
  80. b_3_16·b_5_58 + b_4_30·b_4_31 + b_2_4·b_2_63 + b_2_3·b_2_6·b_4_30
       + b_2_3·b_2_5·b_2_62 + b_2_3·b_2_4·b_2_62 + b_2_32·b_4_32 + b_2_32·b_2_5·b_2_6
       + b_2_3·b_2_5·c_4_35 + b_2_3·b_2_4·c_4_35
  81. b_3_14·b_5_58 + b_4_302 + b_2_3·b_2_52·b_2_6 + b_2_3·b_2_4·b_4_32
       + b_2_3·c_4_35·b_1_22 + b_2_3·c_2_7·b_4_31 + b_2_32·b_2_6·c_2_7
  82. b_3_15·b_5_58 + b_4_30·b_4_31 + b_2_62·b_4_30 + b_2_3·b_2_6·b_4_30
       + b_2_3·b_2_5·b_2_62 + b_2_32·b_4_30 + b_2_32·b_2_5·b_2_6 + b_2_3·b_2_4·c_4_35
  83. b_4_32·b_5_58 + b_2_4·b_2_62·b_3_17 + b_2_4·b_2_5·b_2_6·b_3_16 + b_2_3·b_2_52·b_3_16
       + b_2_3·b_2_4·b_2_5·b_3_16 + b_2_62·c_2_7·b_3_15 + b_2_4·b_2_6·c_4_35·b_1_2
       + b_2_3·c_2_7·c_4_35·b_1_1
  84. b_4_31·b_5_58 + b_2_62·b_5_58 + b_2_3·b_2_62·b_3_17 + b_2_3·b_2_5·b_2_6·b_3_16
       + b_2_32·b_2_5·b_3_16 + b_2_3·b_2_5·c_4_35·b_1_1
  85. b_4_30·b_5_58 + b_2_3·b_2_4·b_2_5·b_3_16 + b_2_3·b_2_6·c_2_7·b_3_16
       + b_2_3·b_2_5·c_4_35·b_1_2
  86. b_5_582 + b_2_3·b_2_5·b_2_6·b_4_32 + b_2_3·b_2_4·b_2_6·b_4_32
       + b_2_32·b_2_52·b_2_6 + b_2_32·b_2_4·b_4_32 + b_2_32·b_2_4·b_2_5·b_2_6
       + b_2_33·b_1_2·b_3_17 + b_2_3·b_2_63·c_2_7 + b_2_3·b_2_52·c_4_35
       + b_2_32·c_2_7·b_4_31 + b_2_33·b_2_6·c_2_7

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 10 and the last generator in degree 5.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_7, a Duflot regular element of degree 2
    2. c_4_35, a Duflot regular element of degree 4
    3. b_1_24 + b_1_12·b_1_22 + b_1_14 + b_2_6·b_1_1·b_1_2 + b_2_62 + b_2_5·b_1_1·b_1_2
         + b_2_52 + c_2_7·b_1_12, an element of degree 4
    4. b_1_12·b_1_24 + b_1_14·b_1_22 + b_2_6·b_1_13·b_1_2 + b_2_62·b_1_1·b_1_2
         + b_2_62·b_1_12 + b_2_5·b_1_1·b_1_23 + b_2_5·b_2_6·b_1_1·b_1_2 + b_2_52·b_1_22
         + b_2_52·b_2_6 + c_2_7·b_1_12·b_1_22 + b_2_6·c_2_7·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_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_6 → 0, an element of degree 2
  8. c_2_7 → c_1_02, an element of degree 2
  9. b_3_14 → 0, an element of degree 3
  10. b_3_15 → 0, an element of degree 3
  11. b_3_16 → 0, an element of degree 3
  12. b_3_17 → 0, an element of degree 3
  13. b_4_30 → 0, an element of degree 4
  14. b_4_31 → 0, an element of degree 4
  15. b_4_32 → 0, an element of degree 4
  16. c_4_35 → c_1_14, an element of degree 4
  17. b_5_58 → 0, an element of degree 5

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_1_2 → c_1_3, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → c_1_3·c_1_4 + c_1_0·c_1_2, an element of degree 2
  7. b_2_6 → c_1_42 + c_1_2·c_1_4, an element of degree 2
  8. c_2_7 → c_1_0·c_1_3 + c_1_02, an element of degree 2
  9. b_3_14 → 0, an element of degree 3
  10. b_3_15 → 0, an element of degree 3
  11. b_3_16 → c_1_43 + c_1_22·c_1_4 + c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  12. b_3_17 → c_1_3·c_1_42 + c_1_2·c_1_3·c_1_4 + c_1_1·c_1_2·c_1_3 + c_1_12·c_1_3 + c_1_0·c_1_42
       + c_1_0·c_1_2·c_1_4, an element of degree 3
  13. b_4_30 → 0, an element of degree 4
  14. b_4_31 → c_1_44 + c_1_22·c_1_42, an element of degree 4
  15. b_4_32 → c_1_1·c_1_2·c_1_3·c_1_4 + c_1_12·c_1_3·c_1_4 + c_1_0·c_1_43 + c_1_0·c_1_22·c_1_4
       + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2, an element of degree 4
  16. c_4_35 → c_1_1·c_1_2·c_1_42 + c_1_1·c_1_22·c_1_4 + c_1_12·c_1_42 + c_1_12·c_1_2·c_1_4
       + c_1_12·c_1_22 + c_1_14, an element of degree 4
  17. b_5_58 → 0, an element of degree 5

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_4, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_3 → c_1_42, an element of degree 2
  5. b_2_4 → c_1_2·c_1_4, an element of degree 2
  6. b_2_5 → c_1_2·c_1_3 + c_1_0·c_1_4, an element of degree 2
  7. b_2_6 → c_1_3·c_1_4 + c_1_32, an element of degree 2
  8. c_2_7 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  9. b_3_14 → c_1_2·c_1_3·c_1_4 + c_1_0·c_1_42, an element of degree 3
  10. b_3_15 → c_1_3·c_1_42 + c_1_32·c_1_4, an element of degree 3
  11. b_3_16 → c_1_43 + c_1_3·c_1_42 + c_1_33 + c_1_1·c_1_42 + c_1_12·c_1_4, an element of degree 3
  12. b_3_17 → c_1_2·c_1_42 + c_1_2·c_1_3·c_1_4 + c_1_2·c_1_32 + c_1_1·c_1_2·c_1_4 + c_1_12·c_1_2
       + c_1_0·c_1_3·c_1_4 + c_1_0·c_1_32, an element of degree 3
  13. b_4_30 → c_1_2·c_1_43 + c_1_2·c_1_3·c_1_42 + c_1_2·c_1_32·c_1_4 + c_1_1·c_1_2·c_1_42
       + c_1_12·c_1_2·c_1_4 + c_1_0·c_1_3·c_1_42 + c_1_0·c_1_32·c_1_4, an element of degree 4
  14. b_4_31 → c_1_44 + c_1_3·c_1_43 + c_1_32·c_1_42 + c_1_33·c_1_4 + c_1_34 + c_1_1·c_1_43
       + c_1_12·c_1_42, an element of degree 4
  15. b_4_32 → c_1_2·c_1_43 + c_1_2·c_1_32·c_1_4 + c_1_1·c_1_2·c_1_42 + c_1_1·c_1_2·c_1_3·c_1_4
       + c_1_12·c_1_2·c_1_4 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_43 + c_1_0·c_1_32·c_1_4
       + c_1_0·c_1_33 + c_1_0·c_1_1·c_1_42 + c_1_0·c_1_12·c_1_4, an element of degree 4
  16. c_4_35 → c_1_32·c_1_42 + c_1_33·c_1_4 + c_1_1·c_1_43 + c_1_1·c_1_3·c_1_42
       + c_1_1·c_1_32·c_1_4 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32 + c_1_14, an element of degree 4
  17. b_5_58 → c_1_2·c_1_3·c_1_43 + c_1_1·c_1_2·c_1_3·c_1_42 + c_1_12·c_1_2·c_1_3·c_1_4
       + c_1_0·c_1_44 + c_1_0·c_1_3·c_1_43 + c_1_0·c_1_33·c_1_4 + c_1_0·c_1_1·c_1_43
       + c_1_0·c_1_12·c_1_42, an element of degree 5

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