Cohomology of group number 654 of order 128

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General information on the group

  • The group has 3 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  −  2·t4  −  t3  −  t2  −  2·t  −  1
    (t  +  1)2 · (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 6:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_1_2, a nilpotent element of degree 1
  4. b_2_3, an element of degree 2
  5. a_3_2, a nilpotent element of degree 3
  6. a_3_3, a nilpotent element of degree 3
  7. a_3_4, a nilpotent element of degree 3
  8. a_4_3, a nilpotent element of degree 4
  9. a_4_4, a nilpotent element of degree 4
  10. b_4_6, an 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_6_8, a nilpotent element of degree 6
  14. a_6_9, a nilpotent element of degree 6

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Ring relations

There are 61 minimal relations of maximal degree 12:

  1. a_1_12 + a_1_02
  2. a_1_0·a_1_2
  3. a_1_22 + a_1_0·a_1_1
  4. a_1_03
  5. b_2_3·a_1_2 + b_2_3·a_1_0
  6. a_1_0·a_3_2
  7. a_1_2·a_3_2 + a_1_0·a_3_3
  8. a_1_2·a_3_3 + a_1_1·a_3_2
  9. a_1_2·a_3_2 + a_1_1·a_3_4 + a_1_1·a_3_3
  10. a_1_2·a_3_2 + a_1_1·a_3_3 + a_1_0·a_3_4
  11. a_1_2·a_3_4 + a_1_1·a_3_2
  12. b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_3·a_3_2 + a_4_3·a_1_1 + a_1_02·a_3_4
  13. a_4_3·a_1_0 + a_1_02·a_3_4
  14. a_4_3·a_1_2 + a_1_02·a_3_4
  15. a_4_4·a_1_1 + a_1_02·a_3_4
  16. b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_3·a_3_2 + a_4_4·a_1_0
  17. b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_3·a_3_2 + a_4_4·a_1_2 + a_1_02·a_3_4
  18. b_4_6·a_1_1 + b_2_3·a_3_4 + b_2_3·a_3_2 + b_2_32·a_1_0 + a_1_02·a_3_4
  19. b_4_6·a_1_0 + b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_32·a_1_1 + a_1_02·a_3_4
  20. b_4_6·a_1_2 + b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_32·a_1_1 + a_1_02·a_3_4
  21. a_3_3·a_3_4 + a_3_32 + a_3_22
  22. a_3_2·a_3_4 + a_3_2·a_3_3
  23. a_3_42
  24. a_3_22 + c_4_7·a_1_0·a_1_1
  25. a_3_32 + c_4_7·a_1_02
  26. a_3_2·a_3_3 + c_4_7·a_1_1·a_1_2
  27. a_4_3·a_3_2 + b_2_3·a_4_3·a_1_1
  28. a_4_3·a_3_4 + a_4_3·a_3_3 + a_4_3·a_3_2
  29. a_4_4·a_3_3 + a_4_3·a_3_2
  30. a_4_4·a_3_2 + a_4_3·a_3_3
  31. a_4_4·a_3_4 + a_4_3·a_3_3 + a_4_3·a_3_2
  32. b_4_6·a_3_3 + b_2_32·a_3_3 + b_2_32·a_3_2 + b_2_33·a_1_0 + a_4_3·a_3_3 + a_4_3·a_3_2
       + b_2_3·c_4_7·a_1_1
  33. b_4_6·a_3_2 + b_2_32·a_3_3 + b_2_32·a_3_2 + b_2_33·a_1_1 + a_4_3·a_3_3 + a_4_3·a_3_2
       + b_2_3·c_4_7·a_1_0
  34. b_4_6·a_3_4 + b_2_33·a_1_1 + b_2_33·a_1_0 + a_4_3·a_3_3 + b_2_3·c_4_7·a_1_1
       + b_2_3·c_4_7·a_1_0
  35. a_6_8·a_1_1 + a_4_3·a_3_3 + a_4_3·a_3_2
  36. a_6_8·a_1_0
  37. a_6_8·a_1_2
  38. a_6_9·a_1_1
  39. a_6_9·a_1_0 + a_4_3·a_3_3
  40. a_6_9·a_1_2 + a_4_3·a_3_3
  41. a_4_32
  42. a_4_3·a_4_4
  43. a_4_42
  44. b_4_62 + b_2_32·b_4_6 + b_2_34 + b_2_32·a_4_4 + b_2_32·a_4_3 + b_2_32·c_4_7
  45. a_4_3·b_4_6 + b_2_3·a_6_8 + b_2_32·a_4_3
  46. a_4_4·b_4_6 + b_2_3·a_6_9
  47. a_6_8·a_3_3 + b_2_32·a_4_3·a_1_1 + a_4_3·c_4_7·a_1_1 + c_4_8·a_1_02·a_3_4
  48. a_6_8·a_3_2 + b_2_3·a_4_3·a_3_3 + b_2_32·a_4_3·a_1_1
  49. a_6_8·a_3_4 + b_2_3·a_4_3·a_3_3 + a_4_3·c_4_7·a_1_1 + c_4_8·a_1_02·a_3_4
       + c_4_7·a_1_02·a_3_4
  50. a_6_9·a_3_3 + b_2_3·a_4_3·a_3_3 + c_4_8·a_1_02·a_3_4 + c_4_7·a_1_02·a_3_4
  51. a_6_9·a_3_2 + b_2_3·a_4_3·a_3_3 + b_2_32·a_4_3·a_1_1 + a_4_3·c_4_7·a_1_1
  52. a_6_9·a_3_4 + b_2_32·a_4_3·a_1_1 + a_4_3·c_4_7·a_1_1 + c_4_8·a_1_02·a_3_4
       + c_4_7·a_1_02·a_3_4
  53. a_4_3·a_6_8
  54. a_4_4·a_6_8
  55. b_4_6·a_6_8 + b_2_33·a_4_3 + b_2_3·a_4_3·c_4_7
  56. a_4_3·a_6_9
  57. a_4_4·a_6_9
  58. b_4_6·a_6_9 + b_2_32·a_6_9 + b_2_33·a_4_4 + b_2_3·a_4_4·c_4_7
  59. a_6_82
  60. a_6_8·a_6_9
  61. a_6_92

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 12.
  • 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_3, 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_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. a_4_3 → 0, an element of degree 4
  9. a_4_4 → 0, an element of degree 4
  10. b_4_6 → 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_6_8 → 0, an element of degree 6
  14. a_6_9 → 0, an element of degree 6

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_1_2 → 0, an element of degree 1
  4. b_2_3 → c_1_22, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. a_3_3 → 0, an element of degree 3
  7. a_3_4 → 0, an element of degree 3
  8. a_4_3 → 0, an element of degree 4
  9. a_4_4 → 0, an element of degree 4
  10. b_4_6 → c_1_02·c_1_22, an element of degree 4
  11. c_4_7 → c_1_24 + c_1_02·c_1_22 + 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_6_8 → 0, an element of degree 6
  14. a_6_9 → 0, 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