Cohomology of group number 630 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 5.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 4 and 5, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 5 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
     − 1
    (t  +  1)2 · (t  −  1)5
  • The a-invariants are -∞,-∞,-∞,-5,-5,-5. 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 4:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. b_2_3, an element of degree 2
  5. b_2_4, an element of degree 2
  6. b_2_5, an element of degree 2
  7. b_2_7, an element of degree 2
  8. c_2_6, a Duflot regular element of degree 2
  9. b_3_13, an element of degree 3
  10. b_3_14, an element of degree 3
  11. b_3_15, an element of degree 3
  12. b_4_21, an element of degree 4
  13. c_4_29, a Duflot regular element of degree 4

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Ring relations

There are 34 minimal relations of maximal degree 8:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_1·b_1_2 + a_1_0·b_1_2
  4. b_2_3·a_1_0
  5. b_2_3·b_1_1
  6. b_2_5·a_1_0 + b_2_4·a_1_0
  7. b_1_13 + b_2_5·b_1_1
  8. b_1_13 + b_2_7·b_1_1 + b_2_4·b_1_1 + b_2_7·a_1_0
  9. b_2_4·b_1_22 + b_2_32
  10. b_2_5·b_1_12 + b_2_52 + b_2_4·b_1_12 + b_2_4·b_2_7 + b_2_42
  11. b_1_2·b_3_13 + b_2_3·b_2_5 + b_2_3·b_2_4
  12. a_1_0·b_3_13
  13. a_1_0·b_3_14
  14. b_1_1·b_3_14 + b_1_1·b_3_13 + b_2_5·b_1_12 + b_2_4·b_1_12
  15. b_1_2·b_3_14 + b_2_3·b_2_7 + a_1_0·b_3_15
  16. b_1_2·b_3_14 + b_1_1·b_3_15 + b_2_3·b_2_7
  17. b_2_4·b_2_5·b_1_2 + b_2_42·b_1_2 + b_2_3·b_3_13
  18. b_2_4·b_2_7·b_1_2 + b_2_3·b_3_14
  19. b_1_12·b_3_13 + b_2_7·b_3_13 + b_2_5·b_3_14 + b_2_5·b_3_13 + b_2_52·b_1_1 + b_2_4·b_3_13
       + b_2_4·b_2_5·b_1_1 + b_2_42·a_1_0
  20. b_1_12·b_3_13 + b_2_5·b_3_13 + b_2_4·b_3_14 + b_2_4·b_3_13 + b_2_4·b_2_5·b_1_1
       + b_2_42·b_1_1 + b_2_42·a_1_0
  21. b_4_21·b_1_2 + b_2_3·b_3_15 + b_2_3·b_2_7·b_1_2 + b_2_3·b_2_4·b_1_2 + a_1_0·b_1_2·b_3_15
       + b_2_7·c_2_6·a_1_0
  22. b_4_21·a_1_0
  23. b_4_21·b_1_1
  24. b_3_13·b_3_14 + b_3_132 + b_2_5·b_1_1·b_3_13 + b_2_4·b_1_1·b_3_13 + b_2_4·b_2_5·b_2_7
       + b_2_4·b_2_52 + b_2_42·b_2_7 + b_2_43
  25. b_3_142 + b_3_132 + b_2_53 + b_2_4·b_2_72 + b_2_4·b_2_5·b_2_7 + b_2_42·b_2_7
       + b_2_42·b_2_5
  26. b_3_152 + b_2_7·b_1_2·b_3_15 + b_2_5·b_1_2·b_3_15 + b_2_5·b_2_7·b_1_22
       + b_2_4·b_1_2·b_3_15 + c_4_29·b_1_22 + b_2_72·c_2_6 + b_2_52·c_2_6 + b_2_42·c_2_6
  27. b_3_132 + b_2_4·b_1_1·b_3_13 + b_2_4·b_2_52 + b_2_43 + c_4_29·b_1_12
  28. b_2_4·b_1_2·b_3_15 + b_2_3·b_4_21 + b_2_32·b_2_7 + b_2_32·b_2_4
  29. b_3_13·b_3_15 + b_2_5·b_4_21 + b_2_4·b_4_21 + b_2_3·b_2_5·b_2_7 + b_2_3·b_2_4·b_2_7
       + b_2_3·b_2_4·b_2_5 + b_2_3·b_2_42
  30. b_3_14·b_3_15 + b_2_7·b_4_21 + b_2_3·b_2_72 + b_2_3·b_2_4·b_2_7 + c_4_29·a_1_0·b_1_2
  31. b_4_21·b_3_13 + b_2_4·b_2_5·b_3_15 + b_2_42·b_3_15 + b_2_3·b_2_5·b_3_14
       + b_2_3·b_2_4·b_3_14 + b_2_3·b_2_4·b_3_13
  32. b_4_21·b_3_15 + b_2_3·b_2_5·b_3_15 + b_2_3·b_2_5·b_2_7·b_1_2
       + b_2_7·a_1_0·b_1_2·b_3_15 + b_2_7·c_2_6·b_3_14 + b_2_5·c_2_6·b_3_13
       + b_2_52·c_2_6·b_1_1 + b_2_4·c_2_6·b_3_13 + b_2_42·c_2_6·b_1_1 + b_2_3·c_4_29·b_1_2
       + c_4_29·a_1_0·b_1_22 + b_2_72·c_2_6·a_1_0
  33. b_4_21·b_3_14 + b_2_4·b_2_7·b_3_15 + b_2_3·b_2_7·b_3_14 + b_2_3·b_2_4·b_3_14
  34. b_4_212 + b_2_3·b_2_7·b_4_21 + b_2_3·b_2_5·b_4_21 + b_2_3·b_2_4·b_4_21
       + b_2_32·b_2_4·b_2_5 + b_2_4·b_2_72·c_2_6 + b_2_4·b_2_52·c_2_6 + b_2_43·c_2_6
       + b_2_32·c_4_29

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 13.
  • However, the last relation was already found in degree 8 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_6, a Duflot regular element of degree 2
    2. c_4_29, a Duflot regular element of degree 4
    3. b_1_24 + b_2_7·b_1_22 + b_2_72 + b_2_5·b_1_22 + b_2_4·b_1_12 + b_2_4·b_2_7
         + b_2_42 + b_2_3·b_2_7 + b_2_3·b_2_5 + b_2_3·b_2_4, an element of degree 4
    4. b_2_7·b_1_24 + b_2_72·b_1_22 + b_2_5·b_1_24 + b_2_4·b_2_72 + b_2_4·b_2_52
         + b_2_42·b_1_12 + b_2_43 + b_2_3·b_2_72 + b_2_3·b_2_4·b_2_5 + b_2_3·b_2_42
         + b_2_32·b_2_7 + b_2_32·b_2_4, an element of degree 6
    5. b_2_3, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 11, 13].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

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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. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_7 → 0, an element of degree 2
  8. c_2_6 → c_1_02, an element of degree 2
  9. b_3_13 → 0, an element of degree 3
  10. b_3_14 → 0, an element of degree 3
  11. b_3_15 → 0, an element of degree 3
  12. b_4_21 → 0, an element of degree 4
  13. c_4_29 → c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_3, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. b_2_5 → c_1_32, an element of degree 2
  7. b_2_7 → c_1_32 + c_1_2·c_1_3 + c_1_22, an element of degree 2
  8. c_2_6 → c_1_0·c_1_3 + c_1_02, an element of degree 2
  9. b_3_13 → c_1_2·c_1_32 + c_1_23 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  10. b_3_14 → c_1_33 + c_1_22·c_1_3 + c_1_23 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  11. b_3_15 → 0, an element of degree 3
  12. b_4_21 → 0, an element of degree 4
  13. c_4_29 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
       + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_3 → c_1_2·c_1_4 + c_1_2·c_1_3, an element of degree 2
  5. b_2_4 → c_1_42 + c_1_32, an element of degree 2
  6. b_2_5 → c_1_42 + c_1_3·c_1_4, an element of degree 2
  7. b_2_7 → c_1_32, an element of degree 2
  8. c_2_6 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  9. b_3_13 → c_1_3·c_1_42 + c_1_33, an element of degree 3
  10. b_3_14 → c_1_32·c_1_4 + c_1_33, an element of degree 3
  11. b_3_15 → c_1_2·c_1_42 + c_1_2·c_1_3·c_1_4 + c_1_1·c_1_2·c_1_4 + c_1_1·c_1_2·c_1_3
       + c_1_12·c_1_2 + c_1_0·c_1_3·c_1_4, an element of degree 3
  12. b_4_21 → c_1_2·c_1_3·c_1_42 + c_1_2·c_1_32·c_1_4 + c_1_1·c_1_2·c_1_42
       + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_2·c_1_4 + c_1_12·c_1_2·c_1_3
       + c_1_0·c_1_3·c_1_42 + c_1_0·c_1_32·c_1_4, an element of degree 4
  13. c_4_29 → c_1_44 + c_1_3·c_1_43 + c_1_32·c_1_42 + c_1_33·c_1_4 + c_1_1·c_1_3·c_1_42
       + c_1_1·c_1_32·c_1_4 + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32
       + c_1_14, an element of degree 4

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