Cohomology of group number 822 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
    ( − 2) · (t3  +  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 12 minimal generators of maximal degree 5:

  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. b_3_5, an element of degree 3
  9. b_4_7, an element of degree 4
  10. c_4_8, a Duflot regular element of degree 4
  11. c_4_9, a Duflot regular element of degree 4
  12. b_5_13, an element of degree 5

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Ring relations

There are 38 minimal relations of maximal degree 10:

  1. a_1_1·a_1_2
  2. a_1_0·a_1_2 + a_1_02
  3. a_1_22 + a_1_12 + a_1_0·a_1_1
  4. a_1_03
  5. b_2_3·a_1_1
  6. a_1_0·a_3_2
  7. a_1_1·a_3_2 + a_1_0·a_3_3 + b_2_3·a_1_02
  8. a_1_2·a_3_3 + a_1_2·a_3_2
  9. a_1_1·a_3_4
  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_3
  12. a_1_1·b_3_5
  13. b_2_3·a_3_3 + b_2_3·a_3_2 + b_2_32·a_1_2 + a_1_02·b_3_5
  14. b_4_7·a_1_1 + a_1_02·a_3_4
  15. b_4_7·a_1_0 + b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_32·a_1_2 + b_2_32·a_1_0
  16. b_4_7·a_1_2 + b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_3·a_3_2 + a_1_02·a_3_4
  17. a_3_3·a_3_4 + a_3_22
  18. a_3_2·a_3_4 + a_3_22
  19. a_3_42 + a_3_32 + a_3_2·a_3_3 + b_2_3·a_1_0·a_3_4
  20. a_3_3·b_3_5 + a_3_2·b_3_5 + b_2_3·a_1_2·b_3_5 + b_2_32·a_1_02
  21. a_3_32 + a_3_22 + c_4_8·a_1_12
  22. a_3_2·a_3_3 + a_3_22 + b_2_3·a_1_0·a_3_4 + c_4_8·a_1_0·a_1_1
  23. a_3_32 + a_3_2·a_3_3 + a_3_22 + c_4_8·a_1_02
  24. b_3_52 + b_2_33 + a_3_3·b_3_5 + a_3_2·b_3_5 + c_4_9·a_1_12 + c_4_9·a_1_0·a_1_1
  25. a_1_1·b_5_13 + a_3_2·a_3_3 + a_3_22 + b_2_3·a_1_0·a_3_4 + c_4_9·a_1_12
  26. b_3_52 + b_2_33 + a_3_4·b_3_5 + a_3_3·b_3_5 + a_1_0·b_5_13 + b_2_3·a_1_0·b_3_5
       + b_2_3·a_1_0·a_3_4 + c_4_9·a_1_12 + c_4_9·a_1_02
  27. b_3_52 + b_2_33 + a_3_4·b_3_5 + a_1_2·b_5_13 + a_3_22
  28. b_4_7·a_3_2 + b_2_3·a_1_02·b_3_5 + b_2_3·c_4_8·a_1_2 + b_2_3·c_4_8·a_1_0
  29. b_4_7·a_3_4 + b_4_7·a_3_3 + b_2_32·a_3_4 + b_2_33·a_1_2 + b_2_3·c_4_8·a_1_0
  30. b_4_7·b_3_5 + b_2_3·b_5_13 + b_4_7·a_3_3 + b_2_32·a_3_2 + b_2_3·c_4_9·a_1_2
  31. b_4_7·a_3_3 + b_2_32·a_3_4 + b_2_33·a_1_2 + a_1_02·b_5_13 + b_2_3·c_4_8·a_1_2
       + b_2_3·c_4_8·a_1_0
  32. b_4_72 + b_2_32·b_4_7 + b_2_32·a_1_2·b_3_5 + b_2_33·a_1_02 + b_2_32·c_4_8
  33. a_3_2·b_5_13 + b_2_32·a_1_0·a_3_4 + b_2_33·a_1_02 + c_4_8·a_1_2·b_3_5
       + c_4_8·a_1_0·b_3_5 + c_4_9·a_1_1·a_3_3 + c_4_9·a_1_0·a_3_4 + c_4_9·a_1_0·a_3_3
       + c_4_8·a_1_1·a_3_3 + c_4_8·a_1_0·a_3_4 + b_2_3·c_4_9·a_1_02
  34. a_3_3·b_5_13 + b_2_3·a_1_2·b_5_13 + c_4_8·a_1_2·b_3_5 + c_4_8·a_1_0·b_3_5
       + c_4_9·a_1_0·a_3_4 + c_4_8·a_1_1·a_3_3 + c_4_8·a_1_0·a_3_4 + c_4_8·a_1_0·a_3_3
       + b_2_3·c_4_8·a_1_02
  35. a_3_4·b_5_13 + b_2_33·a_1_02 + c_4_8·a_1_2·b_3_5 + c_4_9·a_1_1·a_3_3
       + c_4_8·a_1_1·a_3_3 + c_4_8·a_1_0·a_3_4 + b_2_3·c_4_8·a_1_02
  36. b_3_5·b_5_13 + b_2_32·b_4_7 + a_3_3·b_5_13 + b_2_3·a_1_0·b_5_13 + b_2_32·a_1_2·b_3_5
       + b_2_32·a_1_0·b_3_5 + c_4_9·a_1_2·b_3_5 + c_4_9·a_1_1·a_3_3 + c_4_9·a_1_0·a_3_4
       + c_4_8·a_1_1·a_3_3 + c_4_8·a_1_0·a_3_4 + c_4_8·a_1_0·a_3_3 + b_2_3·c_4_9·a_1_02
  37. b_4_7·b_5_13 + b_2_32·b_5_13 + b_2_33·a_3_2 + b_2_34·a_1_2 + b_2_3·a_1_02·b_5_13
       + b_2_32·a_1_02·b_3_5 + b_2_3·c_4_8·b_3_5 + b_2_3·c_4_9·a_3_4 + b_2_3·c_4_8·a_3_2
       + b_2_32·c_4_8·a_1_0 + c_4_9·a_1_02·b_3_5 + c_4_8·a_1_02·b_3_5 + c_4_9·a_1_02·a_3_4
       + c_4_8·a_1_02·a_3_4
  38. b_5_132 + b_2_33·b_4_7 + b_2_32·a_1_2·b_5_13 + b_2_33·a_1_2·b_3_5
       + b_2_33·a_1_0·a_3_4 + b_2_33·c_4_8 + b_2_3·c_4_8·a_1_2·b_3_5
       + b_2_32·c_4_8·a_1_02 + c_4_92·a_1_0·a_1_1 + c_4_8·c_4_9·a_1_12
       + c_4_8·c_4_9·a_1_0·a_1_1 + c_4_82·a_1_12 + c_4_82·a_1_0·a_1_1 + c_4_82·a_1_02

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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_4_8, a Duflot regular element of degree 4
    2. c_4_9, 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].

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. 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. b_3_5 → 0, an element of degree 3
  9. b_4_7 → 0, an element of degree 4
  10. c_4_8 → c_1_04, an element of degree 4
  11. c_4_9 → c_1_14 + c_1_04, an element of degree 4
  12. b_5_13 → 0, an element of degree 5

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. b_3_5 → c_1_23, an element of degree 3
  9. b_4_7 → c_1_02·c_1_22, an element of degree 4
  10. c_4_8 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  11. c_4_9 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  12. b_5_13 → c_1_02·c_1_23, 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