Cohomology of group number 557 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 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.

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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_2, a nilpotent element of degree 1
  3. b_1_1, 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_0, a nilpotent element of degree 3
  8. a_3_3, a nilpotent element of degree 3
  9. a_3_5, a nilpotent element of degree 3
  10. b_3_10, an element of degree 3
  11. a_4_9, a nilpotent 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. a_5_17, 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_2
  3. a_1_0·b_1_1 + a_1_22
  4. a_1_2·b_1_12
  5. a_1_22·b_1_1
  6. a_2_3·a_1_0
  7. b_2_4·a_1_0 + a_2_3·a_1_2
  8. b_2_4·a_1_2 + a_2_3·b_1_1
  9. a_2_32 + c_2_5·a_1_22
  10. b_2_42 + a_2_3·a_1_2·b_1_1 + c_2_5·b_1_12
  11. a_2_3·b_2_4 + c_2_5·a_1_2·b_1_1
  12. a_1_0·a_3_0 + a_2_3·a_1_2·b_1_1
  13. a_1_2·a_3_0
  14. a_1_0·a_3_3 + a_2_3·a_1_2·b_1_1
  15. b_1_1·a_3_0 + a_1_2·a_3_3
  16. a_2_3·b_1_12 + a_1_0·a_3_5
  17. b_1_1·a_3_0 + a_1_2·a_3_5
  18. b_1_1·a_3_0 + a_1_0·b_3_10 + a_2_3·b_1_12
  19. b_1_1·a_3_5 + a_1_2·b_3_10
  20. a_2_3·a_3_0
  21. b_2_4·a_3_0 + a_2_3·a_3_3
  22. b_2_4·a_3_0 + a_2_3·a_3_5
  23. a_1_2·b_1_1·b_3_10
  24. b_2_4·a_3_5 + a_2_3·b_3_10
  25. a_4_9·b_1_1 + b_2_4·a_3_3 + b_2_4·a_3_0 + a_1_22·b_3_10
  26. a_4_9·a_1_0
  27. b_2_4·a_3_0 + a_4_9·a_1_2
  28. b_1_15 + b_4_14·b_1_1 + b_2_4·b_3_10 + b_2_4·a_3_5 + b_2_4·a_3_3 + b_2_4·a_3_0
       + c_2_5·b_1_13
  29. b_4_14·a_1_0 + b_2_4·a_3_0
  30. b_4_14·a_1_2 + b_2_4·a_3_5 + a_1_22·b_3_10
  31. a_3_02
  32. a_3_0·a_3_3
  33. a_3_3·a_3_5 + a_3_32 + a_3_0·a_3_5
  34. a_3_52 + a_3_32
  35. a_3_0·b_3_10 + a_3_3·a_3_5
  36. a_2_3·b_1_1·b_3_10 + a_3_3·a_3_5 + a_3_32
  37. a_3_5·b_3_10 + a_3_3·a_3_5 + a_3_32 + a_2_3·a_1_2·b_3_10 + c_4_16·a_1_2·b_1_1
  38. a_3_32 + a_2_3·a_1_2·b_3_10 + c_4_16·a_1_22
  39. b_2_4·a_4_9 + a_2_3·a_1_2·b_3_10 + c_2_5·b_1_1·a_3_3 + c_2_5·a_1_2·a_3_3
       + c_2_5·a_1_0·a_3_5
  40. a_2_3·a_4_9 + c_2_5·a_1_2·a_3_3 + a_2_3·c_2_5·a_1_2·b_1_1
  41. b_3_102 + b_1_13·b_3_10 + b_4_14·b_1_12 + b_2_4·b_1_1·a_3_3 + c_4_16·b_1_12
       + c_2_5·b_1_14 + c_4_17·a_1_22
  42. b_2_4·b_1_14 + b_2_4·b_4_14 + a_3_3·a_3_5 + a_3_32 + c_2_5·b_1_1·b_3_10
       + b_2_4·c_2_5·b_1_12 + c_2_5·b_1_1·a_3_3 + c_2_5·a_1_2·b_3_10 + c_2_5·a_1_2·a_3_3
       + c_2_5·a_1_0·a_3_5
  43. a_2_3·b_4_14 + a_2_3·a_1_2·b_3_10 + c_2_5·a_1_2·b_3_10 + c_2_5·a_1_0·a_3_5
       + a_2_3·c_2_5·a_1_2·b_1_1
  44. a_3_3·b_3_10 + b_1_1·a_5_17 + b_2_4·b_1_1·a_3_3 + a_3_32 + c_4_17·a_1_2·b_1_1
       + c_2_5·b_1_1·a_3_3 + c_2_5·a_1_2·b_3_10
  45. a_3_3·a_3_5 + a_3_32 + a_1_0·a_5_17 + c_2_5·a_1_0·a_3_5 + a_2_3·c_2_5·a_1_2·b_1_1
  46. a_3_32 + a_1_2·a_5_17 + c_4_17·a_1_22
  47. a_4_9·a_3_3 + a_2_3·c_4_16·a_1_2
  48. a_4_9·a_3_0
  49. a_4_9·a_3_5 + a_2_3·c_4_16·a_1_2
  50. b_1_14·a_3_3 + b_4_14·a_3_3 + a_4_9·b_3_10 + c_2_5·b_1_12·a_3_3 + a_2_3·c_4_16·a_1_2
  51. b_4_14·a_3_0 + a_2_3·c_4_16·a_1_2
  52. b_1_14·b_3_10 + b_4_14·b_3_10 + b_2_4·b_1_12·b_3_10 + b_2_4·b_4_14·b_1_1
       + a_4_9·b_3_10 + c_2_5·b_1_12·b_3_10 + b_2_4·c_4_16·b_1_1 + b_2_4·c_2_5·b_1_13
       + c_2_5·b_1_12·a_3_3 + a_2_3·c_4_16·b_1_1 + c_2_5·a_1_22·b_3_10 + a_2_3·c_4_17·a_1_2
  53. b_4_14·a_3_5 + a_2_3·c_4_16·b_1_1
  54. a_4_9·b_3_10 + b_2_4·a_5_17 + c_2_5·b_1_12·a_3_3 + b_2_4·c_2_5·a_3_3
       + a_2_3·c_4_17·b_1_1 + a_2_3·c_2_5·b_3_10
  55. a_2_3·a_5_17 + c_2_5·a_1_22·b_3_10 + a_2_3·c_4_17·a_1_2 + a_2_3·c_4_16·a_1_2
  56. a_4_92 + a_2_3·c_2_5·a_1_2·b_3_10 + c_2_5·c_4_16·a_1_22
  57. b_4_14·b_1_14 + b_4_142 + b_2_4·b_1_13·b_3_10 + b_2_4·b_1_13·a_3_3 + a_3_3·a_5_17
       + c_2_5·b_1_13·b_3_10 + b_2_4·c_2_5·b_1_1·b_3_10 + c_4_17·a_1_2·a_3_3
       + c_4_16·a_1_2·a_3_3 + c_4_16·a_1_0·a_3_5 + a_2_3·c_2_5·a_1_2·b_3_10
       + c_2_5·c_4_16·b_1_12 + c_2_52·b_1_14 + c_2_5·c_4_17·a_1_22
  58. b_4_14·b_1_14 + b_4_142 + b_2_4·b_1_13·b_3_10 + b_2_4·b_1_13·a_3_3 + a_3_0·a_5_17
       + c_2_5·b_1_13·b_3_10 + b_2_4·c_2_5·b_1_1·b_3_10 + c_4_16·a_1_0·a_3_5
       + a_2_3·c_4_16·a_1_2·b_1_1 + a_2_3·c_2_5·a_1_2·b_3_10 + c_2_5·c_4_16·b_1_12
       + c_2_52·b_1_14 + c_2_5·c_4_17·a_1_22
  59. b_3_10·a_5_17 + b_1_13·a_5_17 + b_4_14·b_1_1·a_3_3 + a_4_9·b_4_14 + b_2_4·b_1_1·a_5_17
       + c_4_17·a_1_2·b_3_10 + c_4_16·b_1_1·a_3_3 + c_2_5·b_1_13·a_3_3 + c_4_17·a_1_0·a_3_5
       + c_4_16·a_1_2·a_3_3 + c_4_16·a_1_0·a_3_5 + a_2_3·c_4_17·a_1_2·b_1_1
       + a_2_3·c_4_16·a_1_2·b_1_1 + c_2_5·c_4_16·a_1_2·b_1_1 + c_2_5·c_4_16·a_1_22
  60. b_4_14·b_1_14 + b_4_142 + b_2_4·b_1_13·b_3_10 + b_2_4·b_1_13·a_3_3 + a_3_5·a_5_17
       + c_2_5·b_1_13·b_3_10 + b_2_4·c_2_5·b_1_1·b_3_10 + c_4_17·a_1_2·a_3_3
       + c_4_16·a_1_2·a_3_3 + a_2_3·c_4_17·a_1_2·b_1_1 + a_2_3·c_2_5·a_1_2·b_3_10
       + c_2_5·c_4_16·b_1_12 + c_2_52·b_1_14 + c_2_5·c_4_17·a_1_22
  61. a_4_9·b_4_14 + b_2_4·b_1_13·a_3_3 + c_2_5·b_1_1·a_5_17 + c_2_5·c_4_17·a_1_2·b_1_1
       + c_2_52·b_1_1·a_3_3 + c_2_52·a_1_2·b_3_10
  62. b_4_14·b_1_14 + b_4_142 + b_2_4·b_1_13·b_3_10 + b_2_4·b_1_13·a_3_3
       + c_2_5·b_1_13·b_3_10 + b_2_4·c_2_5·b_1_1·b_3_10 + c_2_5·a_1_0·a_5_17
       + a_2_3·c_4_16·a_1_2·b_1_1 + a_2_3·c_2_5·a_1_2·b_3_10 + c_2_5·c_4_16·b_1_12
       + c_2_52·b_1_14 + c_2_5·c_4_17·a_1_22 + c_2_52·a_1_0·a_3_5
       + a_2_3·c_2_52·a_1_2·b_1_1
  63. a_4_9·a_5_17 + a_2_3·c_4_17·a_3_3 + a_2_3·c_4_16·a_3_3
  64. b_1_14·a_5_17 + b_4_14·a_5_17 + b_2_4·b_1_12·a_5_17 + b_2_4·b_4_14·a_3_3
       + c_2_5·b_4_14·a_3_3 + b_2_4·c_4_16·a_3_3 + b_2_4·c_2_5·b_1_12·a_3_3
       + a_2_3·c_4_17·b_3_10 + c_4_17·a_1_22·b_3_10 + c_4_16·a_1_22·b_3_10
       + a_2_3·c_4_16·a_3_3 + a_2_3·c_2_5·c_4_16·b_1_1
  65. a_5_172 + c_4_172·a_1_22 + c_4_162·a_1_22

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_12, 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_2 → 0, an element of degree 1
  3. b_1_1 → 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_0 → 0, an element of degree 3
  8. a_3_3 → 0, an element of degree 3
  9. a_3_5 → 0, an element of degree 3
  10. b_3_10 → 0, an element of degree 3
  11. a_4_9 → 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. a_5_17 → 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_2 → 0, an element of degree 1
  3. b_1_1 → c_1_3, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_1·c_1_3, an element of degree 2
  6. c_2_5 → c_1_12, an element of degree 2
  7. a_3_0 → 0, an element of degree 3
  8. a_3_3 → 0, an element of degree 3
  9. a_3_5 → 0, an element of degree 3
  10. b_3_10 → c_1_33 + c_1_1·c_1_32 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  11. a_4_9 → 0, an element of degree 4
  12. b_4_14 → c_1_34 + c_1_1·c_1_33 + c_1_0·c_1_1·c_1_32 + c_1_02·c_1_1·c_1_3, an element of degree 4
  13. c_4_16 → c_1_34 + c_1_0·c_1_33 + c_1_0·c_1_1·c_1_32 + c_1_02·c_1_1·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_0·c_1_1·c_1_32 + c_1_02·c_1_32
       + c_1_02·c_1_1·c_1_3 + c_1_04, an element of degree 4
  15. a_5_17 → 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