Cohomology of group number 757 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 3.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t6  −  2·t3  −  t2  −  t  −  1)
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-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 15 minimal generators of maximal degree 5:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_1_2, an element of degree 1
  4. a_2_3, a nilpotent element of degree 2
  5. b_2_4, an element of degree 2
  6. c_2_5, a Duflot regular element of degree 2
  7. a_3_7, a nilpotent element of degree 3
  8. a_3_8, a nilpotent element of degree 3
  9. b_3_9, an element of degree 3
  10. b_3_10, an element of degree 3
  11. b_4_13, an element of degree 4
  12. b_4_14, an element of degree 4
  13. c_4_16, a Duflot regular element of degree 4
  14. c_4_17, a Duflot regular element of degree 4
  15. b_5_26, 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 65 minimal relations of maximal degree 10:

  1. a_1_12 + a_1_0·a_1_1 + a_1_02
  2. a_1_1·b_1_2
  3. a_1_0·b_1_2
  4. a_1_03
  5. a_1_02·a_1_1
  6. a_2_3·b_1_2
  7. b_2_4·a_1_1 + a_2_3·a_1_0
  8. b_2_4·a_1_0 + a_2_3·a_1_1 + a_2_3·a_1_0
  9. a_2_32 + a_2_3·a_1_0·a_1_1 + c_2_5·a_1_02
  10. a_2_3·b_2_4 + a_2_3·a_1_02 + c_2_5·a_1_0·a_1_1 + c_2_5·a_1_02
  11. b_1_2·a_3_7 + a_2_3·a_1_02
  12. b_1_2·a_3_8 + a_2_32 + a_2_3·a_1_02 + c_2_5·a_1_02
  13. a_1_1·a_3_8 + a_1_1·a_3_7 + a_1_0·a_3_7 + c_2_5·a_1_0·a_1_1 + c_2_5·a_1_02
  14. a_1_1·a_3_7 + a_1_0·a_3_8 + c_2_5·a_1_0·a_1_1
  15. b_1_2·b_3_9 + b_2_42 + a_2_3·a_1_02 + c_2_5·a_1_0·a_1_1
  16. a_1_1·b_3_9 + a_1_0·a_3_7 + a_2_3·a_1_02 + c_2_5·a_1_02
  17. a_1_0·b_3_9 + a_1_1·a_3_7 + a_1_0·a_3_7 + c_2_5·a_1_0·a_1_1 + c_2_5·a_1_02
  18. a_1_1·b_3_10 + a_2_3·a_1_02
  19. a_1_0·b_3_10 + a_2_3·a_1_02
  20. a_1_02·a_3_7
  21. b_2_4·a_3_8 + a_2_3·a_3_7 + a_2_3·c_2_5·a_1_0
  22. b_2_4·a_3_7 + a_2_3·a_3_8 + a_2_3·a_3_7 + a_2_3·c_2_5·a_1_1
  23. b_2_4·a_3_7 + a_2_3·b_3_9 + a_2_3·c_2_5·a_1_1 + a_2_3·c_2_5·a_1_0
  24. a_2_3·b_3_10
  25. b_1_22·b_3_10 + b_4_13·b_1_2 + b_2_4·b_3_9 + b_2_4·a_3_7 + a_2_3·a_3_7
       + a_2_3·c_2_5·a_1_1
  26. b_4_13·a_1_1 + b_2_4·a_3_7 + a_2_3·a_3_7 + a_1_02·a_3_8
  27. b_4_13·a_1_0 + a_2_3·a_3_7 + a_1_02·a_3_8
  28. b_1_22·b_3_10 + b_4_14·b_1_2 + b_2_4·b_3_10 + b_2_4·b_3_9 + b_2_4·a_3_7 + a_2_3·a_3_7
       + a_2_3·c_2_5·a_1_1
  29. b_4_14·a_1_1 + a_2_3·a_3_7 + a_2_3·c_2_5·a_1_1 + a_2_3·c_2_5·a_1_0
  30. b_4_14·a_1_0 + b_2_4·a_3_7 + a_2_3·c_2_5·a_1_1
  31. a_3_82 + a_3_7·a_3_8 + a_3_72 + c_2_5·a_1_0·a_3_8 + c_2_52·a_1_02
  32. a_3_8·b_3_9 + a_3_72 + a_2_3·a_1_0·a_3_7 + a_2_3·c_2_5·a_1_02 + c_2_52·a_1_02
  33. a_3_7·b_3_9 + a_3_7·a_3_8 + a_3_72 + c_2_5·a_1_0·a_3_7
  34. a_3_8·b_3_10 + a_2_3·a_1_0·a_3_8
  35. a_3_7·b_3_10 + a_2_3·c_2_5·a_1_02
  36. b_3_92 + b_2_43 + a_3_82 + a_3_72 + c_4_16·b_1_22 + a_2_3·c_2_5·a_1_02
       + c_2_52·a_1_02
  37. b_3_102 + b_3_92 + a_3_82 + a_3_72 + c_4_17·b_1_22 + c_2_52·a_1_02
  38. a_3_72 + a_2_3·a_1_0·a_3_8 + a_2_3·a_1_0·a_3_7 + c_4_17·a_1_02 + c_4_16·a_1_0·a_1_1
       + c_4_16·a_1_02 + a_2_3·c_2_5·a_1_0·a_1_1 + c_2_52·a_1_02
  39. a_3_82 + a_3_72 + a_2_3·a_1_0·a_3_7 + c_4_17·a_1_0·a_1_1 + c_4_16·a_1_02
       + c_2_52·a_1_02
  40. b_3_92 + b_2_4·b_1_2·b_3_10 + b_2_4·b_4_13 + a_3_7·a_3_8 + c_2_5·a_1_0·a_3_7
       + a_2_3·c_2_5·a_1_0·a_1_1 + a_2_3·c_2_5·a_1_02 + c_2_52·a_1_0·a_1_1
  41. a_2_3·b_4_13 + c_2_5·a_1_0·a_3_7 + a_2_3·c_2_5·a_1_02
  42. b_3_9·b_3_10 + b_3_92 + b_2_4·b_1_2·b_3_10 + b_2_4·b_4_14 + a_3_7·a_3_8
       + a_2_3·a_1_0·a_3_7 + a_2_3·c_2_5·a_1_0·a_1_1 + c_2_52·a_1_0·a_1_1 + c_2_52·a_1_02
  43. a_2_3·b_4_14 + a_3_82 + a_3_7·a_3_8 + a_3_72 + a_2_3·a_1_0·a_3_7 + c_2_5·a_1_0·a_3_7
       + c_2_52·a_1_02
  44. b_3_102 + b_3_9·b_3_10 + b_1_2·b_5_26 + a_2_3·a_1_0·a_3_8 + b_2_42·c_2_5
       + c_2_52·a_1_0·a_1_1
  45. a_1_1·b_5_26 + a_3_7·a_3_8 + a_2_3·a_1_0·a_3_8 + a_2_3·a_1_0·a_3_7 + c_4_16·a_1_0·a_1_1
       + c_2_5·a_1_0·a_3_7 + a_2_3·c_2_5·a_1_0·a_1_1 + a_2_3·c_2_5·a_1_02 + c_2_52·a_1_02
  46. a_1_0·b_5_26 + a_3_82 + a_3_7·a_3_8 + a_2_3·a_1_0·a_3_7 + c_4_16·a_1_02
       + a_2_3·c_2_5·a_1_0·a_1_1 + a_2_3·c_2_5·a_1_02
  47. b_4_13·a_3_8 + a_2_3·c_4_17·a_1_1 + a_2_3·c_4_16·a_1_0 + a_2_3·c_2_5·a_3_8
  48. b_4_13·a_3_7 + a_2_3·c_4_17·a_1_0 + a_2_3·c_4_16·a_1_1 + a_2_3·c_4_16·a_1_0
       + c_2_5·a_1_02·a_3_8 + a_2_3·c_2_52·a_1_0
  49. b_4_13·b_3_9 + b_2_42·b_3_10 + b_2_43·b_1_2 + c_4_16·b_1_23 + b_2_4·c_4_16·b_1_2
       + a_2_3·c_4_17·a_1_1 + a_2_3·c_4_17·a_1_0 + a_2_3·c_4_16·a_1_1 + a_2_3·c_2_5·a_3_8
       + a_2_3·c_2_5·a_3_7 + a_2_3·c_2_52·a_1_0
  50. b_4_14·a_3_8 + a_2_3·c_4_17·a_1_0 + a_2_3·c_4_16·a_1_1 + a_2_3·c_4_16·a_1_0
       + a_2_3·c_2_5·a_3_8 + c_2_5·a_1_02·a_3_8
  51. b_4_14·b_3_10 + b_4_13·b_3_10 + b_2_43·b_1_2 + c_4_16·b_1_23 + b_2_4·c_4_17·b_1_2
       + b_2_4·c_4_16·b_1_2 + c_2_5·a_1_02·a_3_8
  52. b_4_14·a_3_7 + a_2_3·c_4_17·a_1_1 + a_2_3·c_4_17·a_1_0 + a_2_3·c_4_16·a_1_1
       + a_2_3·c_2_5·a_3_8 + a_2_3·c_2_52·a_1_0
  53. b_4_14·b_3_9 + b_4_13·b_3_10 + b_2_42·b_3_10 + c_4_17·b_1_23 + b_2_4·c_4_16·b_1_2
       + a_2_3·c_4_17·a_1_1 + a_2_3·c_4_16·a_1_0 + a_2_3·c_2_5·a_3_8 + a_2_3·c_2_5·a_3_7
       + c_2_5·a_1_02·a_3_8 + a_2_3·c_2_52·a_1_0
  54. b_4_13·b_3_10 + b_2_4·b_5_26 + c_4_17·b_1_23 + b_2_4·c_4_17·b_1_2 + b_2_4·c_4_16·b_1_2
       + b_2_4·c_2_5·b_3_9 + a_2_3·c_4_17·a_1_1 + a_2_3·c_4_17·a_1_0 + a_2_3·c_4_16·a_1_0
       + a_2_3·c_2_5·a_3_8 + a_2_3·c_2_5·a_3_7 + a_2_3·c_2_52·a_1_0
  55. a_2_3·b_5_26 + a_2_3·c_4_17·a_1_0 + a_2_3·c_4_16·a_1_1 + a_2_3·c_2_5·a_3_8
       + c_2_5·a_1_02·a_3_8
  56. b_4_132 + c_4_17·b_1_24 + b_2_4·c_4_16·b_1_22 + b_2_42·c_4_16
       + a_2_3·c_4_17·a_1_0·a_1_1 + a_2_3·c_4_16·a_1_0·a_1_1 + a_2_3·c_2_5·a_1_0·a_3_8
       + a_2_3·c_2_5·a_1_0·a_3_7 + c_2_5·c_4_17·a_1_02 + c_2_5·c_4_16·a_1_02
       + c_2_53·a_1_02
  57. b_4_142 + b_2_44 + c_4_17·b_1_24 + b_2_42·c_4_17 + a_2_3·c_4_16·a_1_0·a_1_1
       + a_2_3·c_2_5·a_1_0·a_3_7 + c_2_5·c_4_16·a_1_02 + a_2_3·c_2_52·a_1_0·a_1_1
       + c_2_53·a_1_02
  58. b_4_13·b_4_14 + b_2_42·b_4_14 + b_2_42·b_4_13 + b_2_44 + c_4_17·b_1_24
       + c_4_16·b_1_2·b_3_10 + b_2_4·c_4_17·b_1_22 + b_2_42·c_4_16
       + a_2_3·c_4_16·a_1_0·a_1_1 + a_2_3·c_4_16·a_1_02 + a_2_3·c_2_5·a_1_0·a_3_7
       + c_2_5·c_4_17·a_1_0·a_1_1 + c_2_5·c_4_17·a_1_02 + c_2_52·a_1_0·a_3_8
       + a_2_3·c_2_52·a_1_02 + c_2_53·a_1_02
  59. a_3_8·b_5_26 + c_4_17·a_1_0·a_3_8 + c_4_16·a_1_0·a_3_8 + c_4_16·a_1_0·a_3_7
       + a_2_3·c_4_17·a_1_02 + a_2_3·c_2_5·a_1_0·a_3_7 + c_2_5·c_4_17·a_1_0·a_1_1
       + c_2_5·c_4_17·a_1_02 + c_2_5·c_4_16·a_1_0·a_1_1 + c_2_5·c_4_16·a_1_02
       + a_2_3·c_2_52·a_1_0·a_1_1 + a_2_3·c_2_52·a_1_02
  60. b_3_10·b_5_26 + b_2_42·b_4_14 + b_2_42·b_4_13 + b_2_44 + c_4_17·b_1_2·b_3_10
       + c_4_16·b_1_2·b_3_10 + b_2_4·c_4_16·b_1_22 + b_2_4·c_2_5·b_4_14 + b_2_4·c_2_5·b_4_13
       + b_2_42·c_4_17 + b_2_42·c_4_16 + a_2_3·c_4_17·a_1_0·a_1_1 + a_2_3·c_4_16·a_1_0·a_1_1
       + a_2_3·c_2_5·a_1_0·a_3_8 + c_2_5·c_4_17·a_1_0·a_1_1 + c_2_5·c_4_16·a_1_0·a_1_1
       + c_2_52·a_1_0·a_3_7 + c_2_53·a_1_02
  61. a_3_7·b_5_26 + c_4_17·a_1_0·a_3_7 + c_4_16·a_1_0·a_3_8 + a_2_3·c_4_17·a_1_0·a_1_1
       + a_2_3·c_4_17·a_1_02 + a_2_3·c_4_16·a_1_0·a_1_1 + a_2_3·c_4_16·a_1_02
       + a_2_3·c_2_5·a_1_0·a_3_8 + c_2_5·c_4_17·a_1_0·a_1_1 + c_2_5·c_4_16·a_1_0·a_1_1
       + c_2_5·c_4_16·a_1_02 + c_2_52·a_1_0·a_3_8
  62. b_3_9·b_5_26 + b_2_42·b_4_14 + b_2_42·b_4_13 + b_2_44 + c_4_16·b_1_2·b_3_10
       + b_2_4·c_4_16·b_1_22 + b_2_42·c_4_17 + b_2_42·c_4_16 + b_2_43·c_2_5
       + c_4_17·a_1_0·a_3_8 + c_4_17·a_1_0·a_3_7 + c_4_16·a_1_0·a_3_7 + a_2_3·c_4_17·a_1_02
       + a_2_3·c_4_16·a_1_02 + a_2_3·c_2_5·a_1_0·a_3_8 + a_2_3·c_2_5·a_1_0·a_3_7
       + c_2_5·c_4_16·b_1_22 + c_2_5·c_4_17·a_1_0·a_1_1 + a_2_3·c_2_52·a_1_0·a_1_1
       + a_2_3·c_2_52·a_1_02
  63. b_4_14·b_5_26 + b_2_43·b_3_10 + b_2_44·b_1_2 + b_4_13·c_4_17·b_1_2
       + b_4_13·c_4_16·b_1_2 + b_2_4·c_4_17·b_3_10 + b_2_4·c_4_17·b_3_9 + b_2_4·c_4_16·b_3_9
       + b_2_4·c_4_16·b_1_23 + b_2_4·c_2_5·b_5_26 + b_2_42·c_4_17·b_1_2
       + b_2_42·c_4_16·b_1_2 + b_2_42·c_2_5·b_3_10 + a_2_3·c_4_17·a_3_7 + a_2_3·c_4_16·a_3_8
       + a_2_3·c_4_16·a_3_7 + b_2_4·c_2_5·c_4_17·b_1_2 + b_2_4·c_2_52·b_3_9
       + a_2_3·c_2_5·c_4_17·a_1_1 + a_2_3·c_2_5·c_4_16·a_1_0 + a_2_3·c_2_52·a_3_7
       + c_2_52·a_1_02·a_3_8 + a_2_3·c_2_53·a_1_0
  64. b_4_13·b_5_26 + b_4_13·c_4_17·b_1_2 + b_2_4·c_4_16·b_3_10 + b_2_4·c_4_16·b_3_9
       + b_2_42·c_4_17·b_1_2 + b_2_42·c_2_5·b_3_10 + b_2_43·c_2_5·b_1_2
       + a_2_3·c_4_17·a_3_7 + c_4_17·a_1_02·a_3_8 + c_4_16·a_1_02·a_3_8
       + c_2_5·c_4_16·b_1_23 + b_2_4·c_2_5·c_4_16·b_1_2 + a_2_3·c_2_5·c_4_17·a_1_1
       + a_2_3·c_2_5·c_4_16·a_1_1 + a_2_3·c_2_5·c_4_16·a_1_0 + a_2_3·c_2_52·a_3_8
       + c_2_52·a_1_02·a_3_8
  65. b_5_262 + b_2_43·c_4_17 + c_4_172·b_1_22 + c_4_16·c_4_17·b_1_22
       + b_2_43·c_2_52 + c_4_172·a_1_02 + c_4_162·a_1_0·a_1_1 + c_4_162·a_1_02
       + a_2_3·c_2_5·c_4_17·a_1_02 + a_2_3·c_2_52·a_1_0·a_3_8 + c_2_52·c_4_16·b_1_22
       + c_2_52·c_4_17·a_1_0·a_1_1 + c_2_52·c_4_17·a_1_02 + c_2_52·c_4_16·a_1_0·a_1_1
       + a_2_3·c_2_53·a_1_0·a_1_1 + a_2_3·c_2_53·a_1_02

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 10.
  • 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_2_5, a Duflot regular element of degree 2
    2. c_4_16, a Duflot regular element of degree 4
    3. c_4_17, a Duflot regular element of degree 4
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 8].
  • 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 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. c_2_5 → c_1_12, an element of degree 2
  7. a_3_7 → 0, an element of degree 3
  8. a_3_8 → 0, an element of degree 3
  9. b_3_9 → 0, an element of degree 3
  10. b_3_10 → 0, an element of degree 3
  11. b_4_13 → 0, an element of degree 4
  12. b_4_14 → 0, an element of degree 4
  13. c_4_16 → c_1_04, an element of degree 4
  14. c_4_17 → c_1_24 + c_1_04, an element of degree 4
  15. b_5_26 → 0, an element of degree 5

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. b_1_2 → c_1_3, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_0·c_1_3, an element of degree 2
  6. c_2_5 → c_1_1·c_1_3 + c_1_12, an element of degree 2
  7. a_3_7 → 0, an element of degree 3
  8. a_3_8 → 0, an element of degree 3
  9. b_3_9 → c_1_02·c_1_3, an element of degree 3
  10. b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32, an element of degree 3
  11. b_4_13 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_0·c_1_33 + c_1_03·c_1_3, an element of degree 4
  12. b_4_14 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_0·c_1_33 + c_1_0·c_1_2·c_1_32
       + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_03·c_1_3, an element of degree 4
  13. c_4_16 → c_1_03·c_1_3 + c_1_04, an element of degree 4
  14. c_4_17 → c_1_22·c_1_32 + c_1_24 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  15. b_5_26 → c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_02·c_1_33 + c_1_02·c_1_2·c_1_32
       + c_1_02·c_1_22·c_1_3 + c_1_02·c_1_1·c_1_32 + c_1_02·c_1_12·c_1_3
       + c_1_03·c_1_32, 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