Cohomology of group number 550 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. a_2_4, a nilpotent element of degree 2
  6. c_2_5, a Duflot regular element of degree 2
  7. a_3_5, 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. a_4_10, a nilpotent element of degree 4
  12. a_4_8, a nilpotent 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

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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. a_2_4·a_1_0 + a_2_3·a_1_2
  8. a_2_3·b_1_1 + a_2_4·a_1_2
  9. a_2_32
  10. a_2_3·a_2_4
  11. a_2_42 + a_2_4·a_1_22 + c_2_5·a_1_22
  12. a_1_0·a_3_5
  13. a_1_2·a_3_5 + a_2_4·a_1_2·b_1_1
  14. a_1_0·a_3_8 + a_2_4·a_1_2·b_1_1
  15. b_1_1·a_3_5 + a_2_4·b_1_12 + a_1_2·a_3_8 + a_2_4·a_1_2·b_1_1
  16. a_1_0·b_3_9 + a_2_4·a_1_2·b_1_1
  17. a_1_2·b_3_9
  18. b_1_1·a_3_5 + a_1_0·b_3_10 + a_2_4·b_1_12 + a_2_4·a_1_2·b_1_1
  19. b_1_1·a_3_8 + b_1_1·a_3_5 + a_1_2·b_3_10
  20. a_2_3·a_3_5
  21. a_2_4·a_3_5 + a_2_3·a_3_8
  22. a_2_3·b_3_9
  23. a_1_2·b_1_1·b_3_10
  24. a_2_3·b_3_10 + a_2_4·a_3_8 + a_2_4·a_3_5
  25. a_4_10·b_1_1 + a_2_4·b_3_9 + a_2_4·b_1_13 + a_1_22·b_3_10 + a_2_4·a_3_8 + a_2_4·a_3_5
  26. a_4_10·a_1_0
  27. a_1_22·b_3_10 + a_4_10·a_1_2 + a_2_4·a_3_5
  28. a_4_8·b_1_1 + a_2_4·b_3_10 + a_2_4·b_3_9
  29. a_4_8·a_1_0 + a_2_4·a_3_5
  30. a_1_22·b_3_10 + a_4_8·a_1_2 + a_2_4·a_3_8 + a_2_4·a_3_5
  31. a_3_52
  32. a_3_5·a_3_8
  33. a_3_8·b_3_9 + a_3_5·b_3_9
  34. b_3_92 + c_2_5·b_1_14
  35. a_3_8·b_3_9 + a_3_5·b_3_10 + a_2_4·b_1_1·b_3_10 + a_2_4·b_1_1·b_3_9 + a_3_82
  36. a_3_8·b_3_9 + a_2_4·b_1_1·b_3_9 + a_2_4·a_1_2·b_3_10
  37. a_3_8·b_3_10 + a_3_5·b_3_10 + a_2_4·a_1_2·a_3_8 + c_4_16·a_1_2·b_1_1
  38. a_3_82 + c_4_16·a_1_22
  39. b_3_102 + b_1_13·b_3_10 + b_1_13·b_3_9 + a_3_8·b_3_9 + a_2_4·b_1_14 + a_3_82
       + a_2_4·a_1_2·a_3_8 + c_4_16·b_1_12 + c_4_17·a_1_22
  40. a_2_3·a_4_10 + a_2_4·a_1_2·a_3_8
  41. a_3_8·b_3_9 + a_2_4·b_1_1·b_3_9 + a_2_4·a_4_10 + a_2_4·a_1_2·a_3_8
       + a_2_4·c_2_5·a_1_2·b_1_1
  42. a_2_3·a_4_8 + a_2_4·a_1_2·a_3_8 + a_2_4·c_2_5·a_1_2·b_1_1
  43. a_3_8·b_3_9 + a_2_4·b_1_1·b_3_9 + a_2_4·a_4_8 + c_2_5·a_1_2·a_3_8
       + a_2_4·c_2_5·a_1_2·b_1_1
  44. b_3_9·b_3_10 + b_1_1·b_5_26 + a_3_5·b_3_10 + c_2_5·b_1_1·b_3_9 + c_2_5·b_1_14
       + c_2_5·a_1_2·b_3_10
  45. a_3_8·b_3_9 + a_1_0·b_5_26 + a_2_4·b_1_1·b_3_9 + a_2_4·a_1_2·a_3_8
  46. a_3_8·b_3_9 + a_1_2·b_5_26 + a_2_4·b_1_1·b_3_9 + a_2_4·a_1_2·a_3_8 + c_2_5·a_1_2·a_3_8
       + a_2_4·c_2_5·a_1_2·b_1_1
  47. a_4_10·a_3_8 + a_2_3·c_4_16·a_1_2
  48. a_4_10·a_3_5
  49. a_4_10·b_3_9 + a_2_4·b_1_12·b_3_9 + a_2_4·c_2_5·b_1_13 + c_2_5·a_4_10·a_1_2
       + a_2_3·c_2_5·a_3_8
  50. a_4_8·a_3_8 + a_2_4·c_4_16·a_1_2 + a_2_3·c_4_16·a_1_2
  51. a_4_8·b_3_10 + a_4_10·b_3_10 + a_4_10·b_3_9 + a_2_4·c_4_16·b_1_1 + a_2_4·c_2_5·b_1_13
       + a_2_4·c_4_16·a_1_2 + a_2_3·c_4_17·a_1_2 + a_2_3·c_4_16·a_1_2
  52. a_4_8·a_3_5 + a_2_3·c_4_16·a_1_2
  53. a_4_8·b_3_9 + a_4_10·b_3_10 + a_4_10·b_3_9 + a_2_4·b_1_12·b_3_10 + a_2_4·b_1_12·b_3_9
       + a_2_4·c_4_16·a_1_2
  54. a_4_10·b_3_9 + a_2_4·b_1_12·b_3_9 + a_2_3·b_5_26 + a_2_4·c_2_5·b_1_13
       + a_2_3·c_2_5·a_3_8
  55. a_4_10·b_3_10 + a_2_4·b_5_26 + a_2_4·b_1_12·b_3_10 + a_2_4·c_2_5·b_3_9
       + a_2_4·c_2_5·b_1_13 + a_2_4·c_4_16·a_1_2 + a_2_4·c_2_5·a_3_8 + a_2_3·c_4_16·a_1_2
       + a_2_3·c_2_5·a_3_8
  56. a_4_102
  57. a_4_82 + a_2_4·c_4_16·a_1_22 + c_2_5·c_4_16·a_1_22
  58. a_4_10·a_4_8 + a_2_4·c_4_16·a_1_2·b_1_1
  59. a_3_8·b_5_26 + a_3_5·b_5_26 + a_2_4·c_4_17·a_1_22 + c_2_5·c_4_16·a_1_22
  60. a_3_8·b_5_26 + a_2_4·b_1_1·b_5_26 + a_2_4·c_4_16·a_1_2·b_1_1 + a_2_4·c_4_17·a_1_22
       + c_2_5·c_4_16·a_1_22
  61. b_3_9·b_5_26 + a_3_8·b_5_26 + c_2_5·b_1_13·b_3_10 + c_2_5·b_1_13·b_3_9
       + a_2_4·c_2_5·b_1_1·b_3_9 + a_2_4·c_2_5·b_1_14 + a_2_4·c_2_5·a_4_10
       + a_2_4·c_2_5·a_1_2·a_3_8 + c_2_52·b_1_14 + c_2_5·c_4_16·a_1_22
       + a_2_4·c_2_52·a_1_2·b_1_1
  62. b_3_10·b_5_26 + b_1_13·b_5_26 + c_4_16·b_1_1·b_3_9 + c_2_5·b_1_1·b_5_26
       + c_2_5·b_1_13·b_3_10 + c_2_5·b_1_13·b_3_9 + a_2_4·c_4_16·b_1_12
       + a_2_4·c_2_5·b_1_1·b_3_10 + a_2_4·c_2_5·b_1_14 + c_4_16·a_1_2·a_3_8
       + a_2_4·c_4_16·a_1_2·b_1_1 + a_2_4·c_2_5·a_4_10 + c_2_52·b_1_1·b_3_9
       + c_2_52·b_1_14 + c_2_5·c_4_16·a_1_2·b_1_1 + c_2_52·a_1_2·b_3_10
       + c_2_5·c_4_16·a_1_22 + a_2_4·c_2_52·a_1_2·b_1_1
  63. a_4_10·b_5_26 + a_2_4·b_1_12·b_5_26 + a_2_4·c_2_5·b_1_12·b_3_10
       + a_2_4·c_2_5·b_1_12·b_3_9 + a_2_4·c_2_52·b_1_13 + c_2_52·a_4_10·a_1_2
       + a_2_3·c_2_5·c_4_16·a_1_2 + a_2_3·c_2_52·a_3_8
  64. a_4_8·b_5_26 + a_4_10·b_5_26 + a_2_4·c_4_16·b_3_9 + a_2_4·c_2_5·b_5_26
       + a_2_4·c_2_5·b_1_12·b_3_10 + a_2_4·c_2_5·b_1_12·b_3_9 + a_2_3·c_4_16·a_3_8
       + a_2_4·c_2_52·b_3_9 + a_2_4·c_2_52·b_1_13 + a_2_4·c_2_5·c_4_16·a_1_2
       + a_2_4·c_2_52·a_3_8 + a_2_3·c_2_52·a_3_8
  65. b_5_262 + c_2_5·b_1_15·b_3_10 + c_2_5·b_1_15·b_3_9 + a_2_4·c_2_5·b_1_13·b_3_9
       + a_2_4·c_2_5·b_1_16 + c_2_5·c_4_16·b_1_14 + c_2_52·b_1_16 + c_2_53·b_1_14
       + c_2_52·c_4_16·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. a_2_4 → 0, an element of degree 2
  6. c_2_5 → c_1_12, an element of degree 2
  7. a_3_5 → 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. a_4_10 → 0, an element of degree 4
  12. a_4_8 → 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, 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_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. a_2_4 → 0, an element of degree 2
  6. c_2_5 → c_1_12, an element of degree 2
  7. a_3_5 → 0, an element of degree 3
  8. a_3_8 → 0, an element of degree 3
  9. b_3_9 → c_1_1·c_1_32, an element of degree 3
  10. b_3_10 → 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_10 → 0, an element of degree 4
  12. a_4_8 → 0, an element of degree 4
  13. c_4_16 → c_1_12·c_1_32 + c_1_0·c_1_33 + c_1_04, an element of degree 4
  14. c_4_17 → c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_0·c_1_33 + c_1_02·c_1_32, an element of degree 4
  15. b_5_26 → c_1_13·c_1_32 + c_1_0·c_1_1·c_1_33 + c_1_02·c_1_1·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