Cohomology of group number 86 of order 128

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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 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t4  +  t3  +  t2  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 14 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. a_2_1, a nilpotent element of degree 2
  4. b_2_2, an element of degree 2
  5. a_3_1, a nilpotent element of degree 3
  6. a_3_2, a nilpotent element of degree 3
  7. a_3_3, a nilpotent element of degree 3
  8. a_3_4, a nilpotent element of degree 3
  9. a_4_4, a nilpotent element of degree 4
  10. a_4_5, a nilpotent element of degree 4
  11. c_4_7, a Duflot regular element of degree 4
  12. c_4_8, a Duflot regular element of degree 4
  13. a_5_10, a nilpotent element of degree 5
  14. a_5_11, a nilpotent element of degree 5

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Ring relations

There are 65 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_13
  4. a_2_1·a_1_1
  5. a_2_1·a_1_0
  6. b_2_2·a_1_0
  7. a_2_12
  8. a_1_1·a_3_1
  9. a_1_0·a_3_1
  10. a_1_0·a_3_2
  11. a_1_1·a_3_3 + a_1_1·a_3_2 + b_2_2·a_1_12
  12. a_1_0·a_3_3
  13. a_1_0·a_3_4
  14. a_2_1·a_3_2 + a_2_1·a_3_1
  15. a_2_1·a_3_1 + a_1_12·a_3_2
  16. a_2_1·a_3_3 + a_2_1·a_3_1
  17. b_2_2·a_3_1 + a_2_1·a_3_4
  18. b_2_2·a_3_1 + a_2_1·a_3_1 + a_1_12·a_3_4
  19. a_4_4·a_1_1
  20. a_4_4·a_1_0 + a_2_1·a_3_1
  21. b_2_2·a_3_2 + b_2_22·a_1_1 + a_4_5·a_1_1 + a_2_1·a_3_1
  22. a_4_5·a_1_0
  23. a_3_12
  24. a_3_22 + b_2_22·a_1_12
  25. a_3_1·a_3_2
  26. a_2_1·b_2_22 + a_3_32 + b_2_22·a_1_12
  27. a_3_1·a_3_3
  28. a_3_1·a_3_4
  29. a_3_3·a_3_4 + a_3_2·a_3_4 + b_2_2·a_1_1·a_3_4
  30. a_2_1·b_2_22 + a_3_42 + a_3_3·a_3_4 + a_3_2·a_3_4 + a_3_2·a_3_3 + c_4_8·a_1_12
  31. a_2_1·a_4_4
  32. a_3_2·a_3_3 + a_2_1·a_4_5
  33. a_3_2·a_3_3 + a_4_5·a_1_12
  34. a_3_2·a_3_4 + a_1_1·a_5_10 + b_2_22·a_1_12
  35. a_1_0·a_5_10
  36. a_1_1·a_5_11 + b_2_22·a_1_12
  37. a_1_0·a_5_11
  38. a_4_4·a_3_2
  39. a_4_4·a_3_1
  40. a_4_4·a_3_4 + a_4_4·a_3_3 + b_2_2·a_1_12·a_3_4
  41. a_4_5·a_3_3 + b_2_2·a_1_12·a_3_4
  42. a_4_5·a_3_1 + a_4_4·a_3_3
  43. a_4_5·a_3_2 + b_2_2·a_4_5·a_1_1 + b_2_2·a_1_12·a_3_4
  44. b_2_2·a_5_10 + b_2_22·a_3_4 + b_2_22·a_3_3 + b_2_23·a_1_1 + a_4_5·a_3_4 + a_4_5·a_3_2
  45. a_4_4·a_3_3 + a_2_1·a_5_10 + b_2_2·a_1_12·a_3_4
  46. a_4_4·a_3_3 + a_1_12·a_5_10 + b_2_2·a_1_12·a_3_4
  47. a_4_4·a_3_3 + a_2_1·a_5_11
  48. a_4_42
  49. a_4_4·a_4_5 + b_2_2·a_4_5·a_1_12
  50. a_4_52 + b_2_2·a_3_32 + b_2_22·a_1_1·a_3_4 + b_2_2·a_4_5·a_1_12
  51. a_3_3·a_5_10 + b_2_2·a_3_32 + a_4_5·a_1_1·a_3_4 + b_2_2·a_4_5·a_1_12
  52. a_3_2·a_5_10 + b_2_22·a_1_1·a_3_4 + b_2_23·a_1_12 + b_2_2·a_4_5·a_1_12
  53. a_3_1·a_5_10
  54. a_3_4·a_5_10 + b_2_2·a_3_32 + b_2_23·a_1_12 + a_4_5·a_1_1·a_3_4 + c_4_8·a_1_1·a_3_2
  55. b_2_22·a_4_4 + a_3_3·a_5_11 + b_2_23·a_1_12
  56. a_3_2·a_5_11 + b_2_23·a_1_12 + b_2_2·a_4_5·a_1_12
  57. a_3_1·a_5_11
  58. a_3_4·a_5_11 + b_2_22·a_1_1·a_3_4 + b_2_2·a_4_5·a_1_12
  59. a_4_4·a_5_10 + b_2_22·a_1_12·a_3_4 + a_4_5·a_1_12·a_3_4
  60. a_4_5·a_5_10 + b_2_2·a_4_5·a_3_4 + b_2_22·a_4_5·a_1_1 + b_2_22·a_1_12·a_3_4
  61. a_4_4·a_5_11 + c_4_8·a_1_12·a_3_2
  62. a_4_5·a_5_11 + b_2_22·a_4_5·a_1_1 + b_2_22·a_1_12·a_3_4 + c_4_8·a_1_12·a_3_2
  63. a_5_102 + b_2_23·a_1_1·a_3_4 + b_2_22·a_4_5·a_1_12 + b_2_22·c_4_8·a_1_12
  64. a_5_112 + b_2_24·a_1_12 + c_4_7·a_3_32
  65. a_5_10·a_5_11 + b_2_2·a_3_3·a_5_11 + b_2_23·a_1_1·a_3_4 + b_2_24·a_1_12
       + b_2_2·a_4_5·a_1_1·a_3_4

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_4_7, a Duflot regular element of degree 4
    2. c_4_8, a Duflot regular element of degree 4
    3. b_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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. a_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. a_3_1 → 0, an element of degree 3
  6. a_3_2 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. a_4_4 → 0, an element of degree 4
  10. a_4_5 → 0, an element of degree 4
  11. c_4_7 → c_1_04, an element of degree 4
  12. c_4_8 → c_1_14 + c_1_04, an element of degree 4
  13. a_5_10 → 0, an element of degree 5
  14. a_5_11 → 0, an element of degree 5

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_22, an element of degree 2
  5. a_3_1 → 0, an element of degree 3
  6. a_3_2 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. a_4_4 → 0, an element of degree 4
  10. a_4_5 → 0, an element of degree 4
  11. c_4_7 → c_1_24 + c_1_04, an element of degree 4
  12. c_4_8 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. a_5_10 → 0, an element of degree 5
  14. a_5_11 → 0, 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