Cohomology of group number 524 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 2.
  • It has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    (2) · (t2  +  1/2)
    (t  +  1) · (t  −  1)4 · (t2  +  1)
  • 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 11 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_14, an element of degree 3
  10. b_3_15, an element of degree 3
  11. c_4_27, a Duflot regular element of degree 4

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Ring relations

There are 27 minimal relations of maximal degree 6:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_1_0·b_1_2
  4. b_2_3·a_1_0
  5. b_2_4·a_1_0
  6. b_2_4·b_1_1 + b_2_3·b_1_2
  7. b_2_5·a_1_0
  8. b_2_7·b_1_1
  9. b_2_32 + c_2_6·b_1_12
  10. b_2_3·b_2_4 + c_2_6·b_1_1·b_1_2
  11. b_2_42 + c_2_6·b_1_22
  12. b_2_5·b_1_22 + b_2_5·b_1_1·b_1_2 + b_2_52
  13. b_1_2·b_3_14 + b_2_4·b_2_5
  14. a_1_0·b_3_14
  15. b_1_1·b_3_14 + b_2_3·b_2_5
  16. b_1_2·b_3_15 + b_2_5·b_2_7
  17. b_2_3·b_2_7 + a_1_0·b_3_15
  18. b_1_1·b_3_15
  19. b_2_3·b_3_14 + b_2_5·c_2_6·b_1_1
  20. b_2_5·b_3_14 + b_2_4·b_2_5·b_1_2 + b_2_3·b_2_5·b_1_2
  21. b_2_4·b_3_14 + b_2_5·c_2_6·b_1_2
  22. b_2_3·b_3_15
  23. b_2_5·b_3_15 + b_2_5·b_2_7·b_1_2
  24. b_2_7·b_3_14 + b_2_4·b_3_15
  25. b_3_142 + b_2_52·c_2_6
  26. b_3_152 + b_2_5·b_2_72 + b_2_7·a_1_0·b_3_15
  27. b_3_14·b_3_15 + b_2_4·b_2_5·b_2_7

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 6.
  • 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_6, a Duflot regular element of degree 2
    2. c_4_27, a Duflot regular element of degree 4
    3. b_1_22 + b_1_1·b_1_2 + b_1_12 + b_2_7, an element of degree 2
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 6].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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_14 → 0, an element of degree 3
  10. b_3_15 → 0, an element of degree 3
  11. c_4_27 → 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_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_2_3 → c_1_0·c_1_2, an element of degree 2
  5. b_2_4 → c_1_0·c_1_3, 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_14 → 0, an element of degree 3
  10. b_3_15 → 0, an element of degree 3
  11. c_4_27 → 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 4

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_2_3 → c_1_0·c_1_2, an element of degree 2
  5. b_2_4 → c_1_0·c_1_3, an element of degree 2
  6. b_2_5 → c_1_32 + c_1_2·c_1_3, 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_14 → c_1_0·c_1_32 + c_1_0·c_1_2·c_1_3, an element of degree 3
  10. b_3_15 → 0, an element of degree 3
  11. c_4_27 → c_1_34 + c_1_2·c_1_33 + 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 4

  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 → 0, an element of degree 2
  5. b_2_4 → c_1_0·c_1_2, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_7 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  8. c_2_6 → c_1_02, an element of degree 2
  9. b_3_14 → 0, an element of degree 3
  10. b_3_15 → 0, an element of degree 3
  11. c_4_27 → 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 4

  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_3, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_0·c_1_3, an element of degree 2
  6. b_2_5 → c_1_32, an element of degree 2
  7. b_2_7 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  8. c_2_6 → c_1_02, an element of degree 2
  9. b_3_14 → c_1_0·c_1_32, an element of degree 3
  10. b_3_15 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  11. c_4_27 → c_1_34 + 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

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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