Cohomology of group number 763 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 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • 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 13 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. b_1_2, 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. b_2_5, an element of degree 2
  7. b_4_7, an element of degree 4
  8. b_4_9, an element of degree 4
  9. a_5_12, a nilpotent element of degree 5
  10. b_5_11, an element of degree 5
  11. b_5_13, an element of degree 5
  12. a_6_18, a nilpotent element of degree 6
  13. c_8_26, a Duflot regular element of degree 8

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Ring relations

There are 47 minimal relations of maximal degree 12:

  1. a_1_12 + a_1_0·a_1_1 + a_1_02
  2. a_1_1·b_1_2
  3. a_1_0·b_1_2
  4. a_1_03
  5. a_2_3·b_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. a_2_32
  9. a_2_3·a_2_4 + a_2_3·a_1_0·a_1_1 + a_2_3·a_1_02
  10. a_2_42 + a_2_3·a_1_0·a_1_1 + a_2_3·a_1_02
  11. b_2_5·a_1_02
  12. b_2_5·a_1_0·a_1_1 + a_2_3·a_1_02
  13. a_2_3·b_2_5·a_1_0
  14. b_4_7·b_1_2 + b_2_52·a_1_0 + a_2_4·b_2_5·b_1_2
  15. b_4_7·a_1_0 + b_2_52·a_1_0 + a_2_3·b_2_5·a_1_1
  16. b_4_9·b_1_2 + b_2_52·a_1_0
  17. b_1_2·a_5_12 + a_2_4·b_2_5·b_1_22
  18. a_1_1·b_5_11 + a_2_3·b_4_7 + a_2_3·b_2_52
  19. a_1_0·b_5_11
  20. b_1_2·b_5_13
  21. a_1_1·b_5_13 + a_2_4·b_4_9 + a_2_4·b_4_7 + a_1_0·a_5_12
  22. a_1_0·b_5_13 + a_2_4·b_4_9 + a_2_4·b_4_7 + a_2_3·b_4_9 + a_2_3·b_4_7 + a_1_1·a_5_12
       + a_1_0·a_5_12
  23. b_2_5·b_4_9·a_1_0 + b_2_53·a_1_0 + a_2_3·b_4_9·a_1_1 + a_2_3·b_2_52·a_1_1
       + a_1_0·a_1_1·a_5_12
  24. b_2_5·b_4_7·a_1_1 + b_2_53·a_1_1 + a_2_3·b_5_11
  25. b_2_5·b_4_9·a_1_1 + b_2_5·b_4_9·a_1_0 + b_2_5·b_4_7·a_1_1 + b_2_53·a_1_0 + a_2_3·b_5_13
       + a_2_4·a_5_12 + a_2_3·b_4_9·a_1_1 + a_2_3·b_2_52·a_1_1
  26. b_2_5·b_4_9·a_1_1 + b_2_5·b_4_7·a_1_1 + a_2_4·b_5_13 + a_2_4·a_5_12 + a_2_3·a_5_12
       + a_2_3·b_4_9·a_1_0
  27. a_6_18·b_1_2 + a_2_4·b_2_52·b_1_2
  28. b_2_5·b_4_9·a_1_0 + b_2_53·a_1_0 + a_6_18·a_1_1 + a_2_4·a_5_12 + a_2_3·b_4_9·a_1_1
       + a_2_3·b_2_52·a_1_1
  29. a_6_18·a_1_0 + a_2_4·a_5_12 + a_2_3·a_5_12 + a_2_3·b_2_52·a_1_1 + a_1_02·a_5_12
  30. b_4_72 + b_2_52·b_4_9
  31. a_2_4·b_2_5·b_4_9 + a_2_4·b_2_5·b_4_7 + a_2_3·b_2_5·b_4_9 + a_2_3·b_2_5·b_4_7
       + b_2_5·a_1_1·a_5_12
  32. b_2_5·a_1_0·a_5_12 + a_2_3·a_1_1·a_5_12 + a_2_3·b_4_9·a_1_02 + a_1_02·a_1_1·a_5_12
  33. a_2_4·b_2_5·b_4_9 + a_2_4·b_2_5·b_4_7 + a_2_3·b_2_5·b_4_9 + a_2_3·b_2_5·b_4_7
       + b_2_5·a_1_0·a_5_12 + a_2_3·a_6_18 + a_2_3·a_1_1·a_5_12 + a_2_3·a_1_0·a_5_12
  34. a_2_4·b_2_5·b_4_9 + a_2_4·b_2_5·b_4_7 + a_2_3·b_2_5·b_4_9 + a_2_3·b_2_5·b_4_7
       + a_2_4·a_6_18 + a_2_3·a_1_1·a_5_12
  35. a_2_4·b_2_5·a_5_12 + a_2_3·b_2_5·a_5_12 + a_2_3·a_1_02·a_5_12
  36. b_4_7·b_5_11 + b_2_52·b_5_13 + b_2_52·a_5_12 + b_2_54·a_1_0 + a_2_4·b_2_5·b_5_11
       + a_2_4·b_2_53·b_1_2 + a_2_3·b_2_5·b_5_11
  37. b_4_9·b_5_11 + b_4_7·b_5_13 + b_4_7·a_5_12 + b_2_54·a_1_0 + a_2_3·b_2_5·a_5_12
       + a_2_3·b_2_53·a_1_1
  38. b_5_11·b_5_13 + b_2_5·b_4_7·b_4_9 + b_2_53·b_4_7 + a_5_12·b_5_11
       + a_2_4·b_2_5·b_1_2·b_5_11 + a_2_4·b_2_52·b_4_7 + a_2_4·b_2_54 + a_2_3·b_2_52·b_4_7
       + a_2_3·b_2_54 + a_2_3·b_2_5·a_6_18
  39. a_5_12·b_5_11 + b_4_7·a_6_18 + b_2_52·a_6_18 + a_2_4·b_2_5·b_1_2·b_5_11
       + a_2_4·b_2_52·b_4_7 + a_2_4·b_2_54 + a_2_3·b_4_7·b_4_9 + a_2_3·b_2_52·b_4_9
       + a_2_3·b_2_52·b_4_7 + a_2_3·b_2_54 + b_4_7·a_1_1·a_5_12
  40. b_5_132 + b_2_5·b_4_92 + b_2_53·b_4_9 + a_5_12·b_5_13 + a_5_12·b_5_11 + b_4_9·a_6_18
       + b_2_52·a_6_18 + a_2_4·b_4_92 + a_2_4·b_2_5·b_1_2·b_5_11 + a_2_4·b_2_54
       + a_2_3·b_2_52·b_4_9 + a_2_3·b_2_54 + a_5_122 + b_4_9·a_1_0·a_5_12
  41. b_5_112 + b_2_52·b_1_2·b_5_11 + b_2_52·b_1_26 + b_2_53·b_4_9 + b_2_55
       + a_2_4·b_1_23·b_5_11 + a_2_4·b_2_5·b_1_26 + c_8_26·b_1_22
  42. b_5_132 + b_5_11·b_5_13 + b_2_5·b_4_92 + b_2_5·b_4_7·b_4_9 + b_2_53·b_4_9
       + b_2_53·b_4_7 + a_5_12·b_5_11 + a_2_4·b_4_92 + a_2_4·b_2_5·b_1_2·b_5_11
       + a_2_4·b_2_54 + a_2_3·b_4_92 + a_2_3·b_2_54 + a_5_122 + b_4_9·a_1_1·a_5_12
       + b_4_7·a_1_1·a_5_12 + c_8_26·a_1_02
  43. b_5_11·b_5_13 + b_2_5·b_4_7·b_4_9 + b_2_53·b_4_7 + a_5_12·b_5_11 + a_2_4·b_4_92
       + a_2_4·b_2_5·b_1_2·b_5_11 + a_2_4·b_2_54 + a_2_3·b_4_92 + a_2_3·b_2_54 + a_5_122
       + b_4_9·a_1_1·a_5_12 + b_4_9·a_1_0·a_5_12 + b_4_7·a_1_1·a_5_12 + c_8_26·a_1_0·a_1_1
  44. a_6_18·b_5_11 + b_2_5·b_4_7·a_5_12 + b_2_53·a_5_12 + a_2_4·b_2_52·b_5_11
       + a_2_4·b_2_54·b_1_2 + a_2_3·b_4_7·b_5_13 + a_2_3·b_2_52·b_5_11 + a_2_3·b_2_54·a_1_1
  45. a_6_18·b_5_13 + b_2_5·b_4_9·a_5_12 + b_2_5·b_4_7·a_5_12 + a_2_3·b_4_9·b_5_13
       + a_2_3·b_4_9·a_5_12 + a_2_3·b_2_52·a_5_12 + a_1_1·a_5_122 + a_1_0·a_5_122
       + a_2_3·c_8_26·a_1_1
  46. a_6_18·b_5_13 + b_2_5·b_4_9·a_5_12 + b_2_5·b_4_7·a_5_12 + a_2_3·b_4_9·b_5_13
       + a_6_18·a_5_12 + a_2_3·b_2_52·a_5_12 + a_1_0·a_5_122 + a_2_3·c_8_26·a_1_0
  47. a_6_182 + a_1_0·a_1_1·a_5_122

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_8_26, a Duflot regular element of degree 8
    2. b_1_24 + b_4_9 + b_4_7 + b_2_52, an element of degree 4
    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 1

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_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. b_4_7 → 0, an element of degree 4
  8. b_4_9 → 0, an element of degree 4
  9. a_5_12 → 0, an element of degree 5
  10. b_5_11 → 0, an element of degree 5
  11. b_5_13 → 0, an element of degree 5
  12. a_6_18 → 0, an element of degree 6
  13. c_8_26 → c_1_08, 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. b_1_2 → c_1_1, 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 + c_1_1·c_1_2, an element of degree 2
  7. b_4_7 → 0, an element of degree 4
  8. b_4_9 → 0, an element of degree 4
  9. a_5_12 → 0, an element of degree 5
  10. b_5_11 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_02·c_1_13
       + c_1_04·c_1_1, an element of degree 5
  11. b_5_13 → 0, an element of degree 5
  12. a_6_18 → 0, an element of degree 6
  13. c_8_26 → c_1_14·c_1_24 + c_1_16·c_1_22 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22
       + c_1_04·c_1_14 + c_1_08, 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. b_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. b_4_7 → c_1_12·c_1_22, an element of degree 4
  8. b_4_9 → c_1_14, an element of degree 4
  9. a_5_12 → 0, an element of degree 5
  10. b_5_11 → c_1_25 + c_1_12·c_1_23, an element of degree 5
  11. b_5_13 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  12. a_6_18 → 0, an element of degree 6
  13. c_8_26 → c_1_12·c_1_26 + c_1_16·c_1_22 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22
       + c_1_04·c_1_14 + c_1_08, 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