Cohomology of group number 40 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 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
    t5  −  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 4:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_0, a nilpotent element of degree 2
  4. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. a_3_1, a nilpotent element of degree 3
  7. a_3_2, a nilpotent element of degree 3
  8. a_3_4, a nilpotent element of degree 3
  9. b_3_3, an element of degree 3
  10. a_4_2, a nilpotent element of degree 4
  11. a_4_3, a nilpotent element of degree 4
  12. a_4_5, a nilpotent element of degree 4
  13. c_4_6, a Duflot regular element of degree 4
  14. c_4_7, a Duflot regular element of degree 4

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Ring relations

There are 65 minimal relations of maximal degree 8:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_0
  5. a_2_1·a_1_1
  6. a_2_1·a_1_0 + a_2_0·a_1_1
  7. b_2_2·a_1_1
  8. b_2_2·a_1_0 + a_2_0·a_1_1
  9. a_2_02
  10. a_2_0·a_2_1
  11. a_2_12
  12. a_2_0·b_2_2 + a_1_1·a_3_1
  13. a_1_0·a_3_1
  14. a_2_1·b_2_2 + a_2_0·b_2_2 + a_1_1·a_3_2
  15. a_2_0·b_2_2 + a_1_0·a_3_2
  16. a_1_1·a_3_4
  17. a_2_1·b_2_2 + a_1_0·a_3_4
  18. a_1_1·b_3_3
  19. a_1_0·b_3_3 + a_2_0·b_2_2
  20. a_2_0·a_3_1
  21. a_2_1·a_3_1 + a_2_0·a_3_2
  22. b_2_2·a_3_2 + b_2_2·a_3_1 + a_2_1·a_3_2 + a_2_1·a_3_1
  23. b_2_2·a_3_2 + b_2_2·a_3_1 + a_2_0·a_3_4
  24. a_2_1·a_3_4
  25. a_2_0·b_3_3
  26. b_2_2·a_3_2 + b_2_2·a_3_1 + a_2_1·b_3_3
  27. a_4_2·a_1_1 + a_2_1·a_3_1
  28. a_4_2·a_1_0
  29. b_2_2·a_3_2 + b_2_2·a_3_1 + a_4_3·a_1_1 + a_2_1·a_3_1
  30. a_4_3·a_1_0 + a_2_1·a_3_1
  31. a_4_5·a_1_1
  32. b_2_2·a_3_2 + b_2_2·a_3_1 + a_4_5·a_1_0
  33. a_3_12
  34. a_3_22
  35. a_3_1·a_3_2
  36. a_3_2·a_3_4 + a_3_1·a_3_4
  37. a_3_42
  38. a_3_2·b_3_3 + a_3_1·b_3_3
  39. b_3_32 + b_2_23
  40. a_3_2·b_3_3 + b_2_2·a_4_2
  41. a_2_0·a_4_2
  42. a_2_1·a_4_2
  43. a_3_2·b_3_3 + b_2_2·a_4_3 + a_3_2·a_3_4
  44. a_2_0·a_4_3
  45. a_2_1·a_4_3
  46. a_3_4·b_3_3 + b_2_2·a_4_5
  47. a_2_0·a_4_5
  48. a_2_1·a_4_5
  49. a_4_2·a_3_2 + a_2_0·c_4_7·a_1_1 + a_2_0·c_4_6·a_1_1
  50. a_4_2·a_3_1
  51. a_4_2·b_3_3 + b_2_22·a_3_1
  52. a_4_3·a_3_2
  53. a_4_3·a_3_1 + a_2_0·c_4_7·a_1_1 + a_2_0·c_4_6·a_1_1
  54. a_4_3·a_3_4 + a_4_2·a_3_4 + a_2_0·c_4_6·a_1_1
  55. a_4_3·b_3_3 + b_2_22·a_3_1 + a_4_2·a_3_4
  56. a_4_5·a_3_2 + a_4_2·a_3_4 + a_2_0·c_4_6·a_1_1
  57. a_4_5·a_3_1 + a_4_2·a_3_4
  58. a_4_5·a_3_4
  59. a_4_5·b_3_3 + b_2_22·a_3_4 + a_2_0·c_4_6·a_1_1
  60. a_4_22
  61. a_4_32
  62. a_4_2·a_4_3
  63. a_4_3·a_4_5 + b_2_2·a_3_1·a_3_4
  64. a_4_52
  65. a_4_2·a_4_5 + b_2_2·a_3_1·a_3_4


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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_6, a Duflot regular element of degree 4
    2. c_4_7, 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_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. a_2_00, an element of degree 2
  4. a_2_10, an element of degree 2
  5. b_2_20, an element of degree 2
  6. a_3_10, an element of degree 3
  7. a_3_20, an element of degree 3
  8. a_3_40, an element of degree 3
  9. b_3_30, an element of degree 3
  10. a_4_20, an element of degree 4
  11. a_4_30, an element of degree 4
  12. a_4_50, an element of degree 4
  13. c_4_6c_1_14, an element of degree 4
  14. c_4_7c_1_14 + c_1_04, an element of degree 4

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

  1. a_1_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. a_2_00, an element of degree 2
  4. a_2_10, an element of degree 2
  5. b_2_2c_1_22, an element of degree 2
  6. a_3_10, an element of degree 3
  7. a_3_20, an element of degree 3
  8. a_3_40, an element of degree 3
  9. b_3_3c_1_23, an element of degree 3
  10. a_4_20, an element of degree 4
  11. a_4_30, an element of degree 4
  12. a_4_50, an element of degree 4
  13. c_4_6c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. c_4_7c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4


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