Cohomology of group number 760 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 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
    ( − 2) · (t2  +  1/2·t  +  1/2)
    (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 15 minimal generators of maximal degree 8:

  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. a_2_3, a nilpotent element of degree 2
  5. a_2_4, a nilpotent element of degree 2
  6. b_2_5, an element of degree 2
  7. a_3_7, a nilpotent element of degree 3
  8. b_4_8, an element of degree 4
  9. c_4_9, a Duflot regular element of degree 4
  10. a_5_11, a nilpotent element of degree 5
  11. a_5_12, a nilpotent element of degree 5
  12. a_5_13, a nilpotent element of degree 5
  13. a_6_16, a nilpotent element of degree 6
  14. a_7_23, a nilpotent element of degree 7
  15. c_8_26, a Duflot regular element of degree 8

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Ring relations

There are 62 minimal relations of maximal degree 14:

  1. a_1_12 + a_1_0·a_1_1 + a_1_02
  2. a_1_1·a_1_2
  3. a_1_0·a_1_2
  4. a_1_03
  5. a_2_3·a_1_2
  6. a_2_4·a_1_1 + a_2_3·a_1_0
  7. a_2_4·a_1_0 + a_2_3·a_1_1 + a_2_3·a_1_0
  8. b_2_5·a_1_0 + a_1_23
  9. a_2_32 + a_2_3·a_1_02
  10. a_2_3·a_2_4 + a_2_3·a_1_0·a_1_1 + a_2_3·a_1_02
  11. a_1_1·a_3_7 + a_2_3·a_1_02
  12. a_1_0·a_3_7
  13. a_1_2·a_3_7 + a_2_42 + a_2_3·a_1_0·a_1_1
  14. b_2_5·a_1_23
  15. a_2_3·a_3_7
  16. b_4_8·a_1_0 + a_2_42·a_1_2
  17. b_4_8·a_1_2 + b_2_5·a_3_7 + b_2_52·a_1_2
  18. a_3_72 + c_4_9·a_1_22
  19. a_1_2·a_5_11 + b_2_52·a_1_22 + a_2_42·b_2_5
  20. a_1_1·a_5_12 + a_1_0·a_5_11 + a_2_4·b_2_5·a_1_22
  21. a_1_1·a_5_11 + a_1_0·a_5_12 + a_1_0·a_5_11
  22. a_1_2·a_5_12
  23. a_2_3·b_2_52 + a_1_1·a_5_13 + a_1_0·a_5_11 + a_2_4·b_2_5·a_1_22
  24. a_1_1·a_5_11 + a_1_0·a_5_13 + a_1_0·a_5_11
  25. b_4_8·a_3_7 + b_2_52·a_3_7 + a_2_42·b_2_5·a_1_2 + b_2_5·c_4_9·a_1_2
  26. a_1_02·a_5_11 + a_2_42·b_2_5·a_1_2
  27. a_2_4·a_5_11 + a_2_4·b_2_5·a_3_7 + a_2_4·b_2_52·a_1_2 + a_2_3·a_5_12
  28. a_2_4·a_5_12 + a_2_4·a_5_11 + a_2_4·b_2_5·a_3_7 + a_2_4·b_2_52·a_1_2 + a_2_3·a_5_11
       + a_2_43·a_1_2
  29. b_2_5·a_5_11 + b_2_5·b_4_8·a_1_1 + b_2_52·a_3_7 + b_2_53·a_1_2 + a_1_22·a_5_13
       + a_2_42·b_2_5·a_1_2 + a_2_43·a_1_2
  30. a_2_4·a_5_11 + a_2_4·b_2_5·a_3_7 + a_2_4·b_2_52·a_1_2 + a_2_3·a_5_13 + a_2_43·a_1_2
  31. a_6_16·a_1_1 + a_2_4·a_5_11 + a_2_4·b_2_5·a_3_7 + a_2_4·b_2_52·a_1_2 + a_2_3·a_5_11
       + a_1_02·a_5_12 + a_2_42·b_2_5·a_1_2 + a_2_43·a_1_2 + a_2_3·c_4_9·a_1_1
  32. a_6_16·a_1_0 + a_2_3·a_5_11 + a_2_42·b_2_5·a_1_2 + a_2_43·a_1_2 + a_2_3·c_4_9·a_1_0
  33. a_6_16·a_1_2 + a_2_4·b_2_52·a_1_2 + a_2_42·b_2_5·a_1_2 + a_2_43·a_1_2
       + c_4_9·a_1_23
  34. b_4_82 + b_2_54 + b_2_52·c_4_9
  35. a_3_7·a_5_11 + a_2_42·b_2_52 + b_2_5·c_4_9·a_1_22
  36. a_3_7·a_5_12 + a_2_3·a_1_0·a_5_11
  37. a_2_3·a_6_16 + a_2_3·c_4_9·a_1_02
  38. a_2_4·a_6_16 + a_2_42·b_2_52 + a_2_43·b_2_5 + a_2_4·c_4_9·a_1_22
       + a_2_3·c_4_9·a_1_0·a_1_1 + a_2_3·c_4_9·a_1_02
  39. a_2_3·b_2_5·b_4_8 + a_1_1·a_7_23 + a_2_3·a_1_0·a_5_12 + a_2_3·a_1_0·a_5_11
       + a_2_3·c_4_9·a_1_0·a_1_1 + a_2_3·c_4_9·a_1_02
  40. a_1_0·a_7_23 + a_2_3·a_1_0·a_5_12 + a_2_3·a_1_0·a_5_11 + a_2_3·c_4_9·a_1_02
  41. a_3_7·a_5_13 + a_1_2·a_7_23 + b_2_5·a_1_2·a_5_13 + a_2_42·b_2_52 + a_2_4·a_1_2·a_5_13
       + a_2_43·b_2_5 + a_2_3·a_1_0·a_5_11 + a_2_42·c_4_9 + a_2_3·c_4_9·a_1_0·a_1_1
  42. b_4_8·a_5_11 + b_2_54·a_1_2 + b_2_54·a_1_1 + b_2_5·a_1_22·a_5_13 + a_2_42·a_5_13
       + a_2_3·a_1_02·a_5_12 + b_2_52·c_4_9·a_1_2 + b_2_52·c_4_9·a_1_1
       + a_2_4·c_4_9·a_1_23
  43. a_6_16·a_3_7 + a_2_4·b_2_52·a_3_7 + a_2_42·c_4_9·a_1_2 + a_2_4·c_4_9·a_1_23
  44. b_4_8·a_5_13 + b_4_8·a_5_12 + b_2_5·a_7_23 + b_2_53·a_3_7 + a_2_4·b_2_5·a_5_13
       + a_2_4·b_2_52·a_3_7 + b_2_5·c_4_9·a_3_7
  45. a_2_3·a_7_23 + a_2_3·a_1_02·a_5_12 + a_2_4·c_4_9·a_1_23
  46. a_5_122 + a_5_11·a_5_12 + a_5_112 + b_2_54·a_1_22 + a_2_4·b_2_53·a_1_22
       + a_2_43·b_2_52 + b_2_52·c_4_9·a_1_22
  47. b_2_52·a_6_16 + a_2_4·b_2_54 + a_5_12·a_5_13 + a_5_122 + a_2_42·b_2_53
       + c_4_9·a_1_1·a_5_13 + c_4_9·a_1_0·a_5_11 + b_2_52·c_4_9·a_1_22
       + a_2_4·b_2_5·c_4_9·a_1_22
  48. a_5_122 + a_5_11·a_5_13 + a_5_112 + b_2_5·a_1_2·a_7_23 + b_2_5·a_1_1·a_7_23
       + b_2_54·a_1_22 + a_2_42·b_2_53 + a_2_4·b_2_5·a_1_2·a_5_13
       + a_2_4·b_2_53·a_1_22 + b_2_52·c_4_9·a_1_22 + a_2_42·b_2_5·c_4_9
  49. a_5_122 + a_5_11·a_5_13 + a_5_112 + a_3_7·a_7_23 + b_2_5·a_1_1·a_7_23
       + b_2_52·a_1_2·a_5_13 + b_2_54·a_1_22 + a_2_4·a_1_2·a_7_23
       + a_2_4·b_2_5·a_1_2·a_5_13 + a_2_4·b_2_53·a_1_22 + c_4_9·a_1_2·a_5_13
       + a_2_4·b_2_5·c_4_9·a_1_22 + a_2_43·c_4_9 + c_4_92·a_1_22
  50. a_5_122 + c_8_26·a_1_0·a_1_1
  51. a_5_112 + b_2_54·a_1_22 + c_8_26·a_1_02 + b_2_52·c_4_9·a_1_22
  52. a_5_132 + a_5_122 + b_2_52·a_1_2·a_5_13 + b_2_52·a_1_1·a_5_13 + a_2_43·b_2_52
       + c_8_26·a_1_22 + b_2_52·c_4_9·a_1_22 + a_2_42·b_2_5·c_4_9
       + a_2_4·b_2_5·c_4_9·a_1_22
  53. a_6_16·a_5_13 + a_6_16·a_5_12 + a_2_4·b_2_52·a_5_13 + a_2_42·b_2_5·a_5_13
       + c_4_9·a_1_22·a_5_13 + a_2_43·c_4_9·a_1_2
  54. b_4_8·a_7_23 + b_2_53·a_5_13 + b_2_53·a_5_12 + b_2_54·a_3_7 + a_2_4·b_2_5·a_7_23
       + a_2_42·b_2_5·a_5_13 + a_2_4·b_2_5·a_1_22·a_5_13 + a_2_43·a_5_13
       + b_2_5·c_4_9·a_5_13 + b_2_5·c_4_9·a_5_12 + b_2_52·c_4_9·a_3_7 + b_2_53·c_4_9·a_1_2
       + a_2_4·b_2_5·c_4_9·a_3_7 + a_2_4·b_2_52·c_4_9·a_1_2 + a_2_42·b_2_5·c_4_9·a_1_2
       + b_2_5·c_4_92·a_1_2
  55. a_6_16·a_5_12 + a_6_16·a_5_11 + a_2_4·b_2_53·a_3_7 + a_2_4·b_2_54·a_1_2
       + a_2_4·b_2_5·a_1_22·a_5_13 + a_2_43·a_5_13 + a_2_3·c_8_26·a_1_1 + a_2_3·c_4_9·a_5_12
       + a_2_3·c_4_9·a_5_11 + c_8_26·a_1_02·a_1_1 + a_2_42·b_2_5·c_4_9·a_1_2
       + a_2_43·c_4_9·a_1_2
  56. a_6_16·a_5_11 + a_2_4·b_2_53·a_3_7 + a_2_4·b_2_54·a_1_2 + a_2_43·a_5_13
       + a_2_3·c_8_26·a_1_0 + a_2_3·c_4_9·a_5_11 + a_2_42·b_2_5·c_4_9·a_1_2
  57. b_2_5·b_4_8·a_6_16 + a_2_4·b_2_53·b_4_8 + a_5_12·a_7_23 + a_2_42·b_2_54
       + c_4_9·a_1_1·a_7_23 + a_2_42·b_2_52·c_4_9 + a_2_3·c_4_92·a_1_0·a_1_1
       + a_2_3·c_4_92·a_1_02
  58. a_5_11·a_7_23 + b_2_53·a_1_2·a_5_13 + b_2_53·a_1_1·a_5_13 + a_2_42·b_2_54
       + a_2_4·b_2_5·a_1_2·a_7_23 + a_2_4·b_2_54·a_1_22 + b_2_5·c_4_9·a_1_2·a_5_13
       + b_2_5·c_4_9·a_1_1·a_5_13 + b_2_53·c_4_9·a_1_22 + a_2_42·b_2_52·c_4_9
       + a_2_43·b_2_5·c_4_9 + a_2_3·c_8_26·a_1_0·a_1_1 + a_2_3·c_4_9·a_1_0·a_5_11
       + b_2_5·c_4_92·a_1_22
  59. a_6_162 + a_2_42·b_2_54 + a_2_3·c_8_26·a_1_02 + a_2_3·c_4_92·a_1_02
  60. a_5_13·a_7_23 + a_5_12·a_7_23 + a_5_11·a_7_23 + b_2_52·a_1_1·a_7_23
       + b_2_53·a_1_1·a_5_13 + a_2_42·b_2_54 + a_2_43·b_2_53 + c_4_9·a_1_2·a_7_23
       + b_2_5·c_8_26·a_1_22 + b_2_5·c_4_9·a_1_1·a_5_13 + a_2_42·c_8_26
       + a_2_4·c_8_26·a_1_22 + a_2_4·c_4_9·a_1_2·a_5_13 + a_2_4·b_2_52·c_4_9·a_1_22
       + a_2_43·b_2_5·c_4_9 + a_2_3·c_4_9·a_1_0·a_5_11 + a_2_42·c_4_92
       + a_2_3·c_4_92·a_1_0·a_1_1
  61. a_6_16·a_7_23 + a_2_4·b_2_52·a_7_23 + a_2_42·b_2_52·a_5_13
       + a_2_4·b_2_52·a_1_22·a_5_13 + a_2_42·c_4_9·a_5_13 + a_2_4·c_4_9·a_1_22·a_5_13
       + a_2_3·c_4_9·a_1_02·a_5_12 + a_2_42·c_4_92·a_1_2
  62. a_7_232 + b_2_54·a_1_2·a_5_13 + b_2_54·a_1_1·a_5_13 + a_2_43·b_2_54
       + b_2_52·c_8_26·a_1_22 + b_2_52·c_4_9·a_1_2·a_5_13 + b_2_52·c_4_9·a_1_1·a_5_13
       + a_2_42·b_2_53·c_4_9 + a_2_4·b_2_53·c_4_9·a_1_22 + a_2_43·b_2_52·c_4_9
       + c_4_9·c_8_26·a_1_22 + b_2_52·c_4_92·a_1_22 + a_2_42·b_2_5·c_4_92
       + a_2_4·b_2_5·c_4_92·a_1_22 + c_4_93·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 14.
  • 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_9, a Duflot regular element of degree 4
    2. c_8_26, a Duflot regular element of degree 8
    3. b_2_5, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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 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. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. a_3_7 → 0, an element of degree 3
  8. b_4_8 → 0, an element of degree 4
  9. c_4_9 → c_1_04, an element of degree 4
  10. a_5_11 → 0, an element of degree 5
  11. a_5_12 → 0, an element of degree 5
  12. a_5_13 → 0, an element of degree 5
  13. a_6_16 → 0, an element of degree 6
  14. a_7_23 → 0, an element of degree 7
  15. c_8_26 → c_1_18, an element of degree 8

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. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → c_1_22, an element of degree 2
  7. a_3_7 → 0, an element of degree 3
  8. b_4_8 → c_1_24 + c_1_02·c_1_22, an element of degree 4
  9. c_4_9 → c_1_04, an element of degree 4
  10. a_5_11 → 0, an element of degree 5
  11. a_5_12 → 0, an element of degree 5
  12. a_5_13 → 0, an element of degree 5
  13. a_6_16 → 0, an element of degree 6
  14. a_7_23 → 0, an element of degree 7
  15. c_8_26 → c_1_28 + c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_26 + c_1_06·c_1_22, an element of degree 8

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