Cohomology of group number 1632 of order 128

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General information on the group

  • The group has 4 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has 2 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
    t5  +  t2  +  1
    (t  −  1)4 · (t2  +  1) · (t4  +  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 8:

  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. c_1_3, a Duflot regular element of degree 1
  5. a_2_8, a nilpotent element of degree 2
  6. b_2_9, an element of degree 2
  7. b_3_19, an element of degree 3
  8. a_5_47, a nilpotent element of degree 5
  9. b_5_52, an element of degree 5
  10. a_6_65, a nilpotent element of degree 6
  11. c_8_146, a Duflot regular element of degree 8

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Ring relations

There are 27 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_8·a_1_0
  4. b_2_9·b_1_1 + a_1_0·b_1_22 + b_2_9·a_1_0
  5. a_2_82
  6. a_1_0·b_3_19 + b_2_9·a_1_0·b_1_2 + a_2_8·b_2_9
  7. b_1_1·b_3_19 + a_1_0·b_1_23 + b_2_9·a_1_0·b_1_2 + a_2_8·b_1_22 + a_2_8·b_2_9
  8. a_1_0·b_1_24 + b_2_9·a_1_0·b_1_22
  9. a_1_0·b_1_24 + a_2_8·b_3_19 + a_2_8·b_2_9·b_1_2
  10. b_3_192 + b_2_9·b_1_24 + b_2_92·b_1_22 + b_2_92·a_1_0·b_1_2
       + a_2_8·b_2_9·b_1_22
  11. a_2_8·b_2_9·b_1_22 + a_2_8·b_2_92 + a_1_0·a_5_47
  12. b_1_1·a_5_47 + a_2_8·b_2_9·b_1_22 + a_2_8·b_2_92
  13. a_1_0·b_5_52 + b_2_92·a_1_0·b_1_2 + a_2_8·b_2_92
  14. a_2_8·a_5_47
  15. b_2_9·b_5_52 + b_2_92·b_3_19 + b_1_22·a_5_47 + b_2_93·a_1_0
  16. a_1_0·b_1_2·a_5_47 + a_6_65·a_1_0
  17. a_6_65·b_1_1 + a_2_8·b_5_52 + a_2_8·b_1_1·b_1_24 + a_2_8·b_1_12·b_1_23
       + a_2_8·b_2_9·b_3_19 + a_1_0·b_1_2·a_5_47
  18. b_3_19·a_5_47 + b_2_9·a_6_65 + a_2_8·b_2_9·b_1_2·b_3_19
  19. b_3_19·b_5_52 + b_2_92·b_1_24 + b_2_93·b_1_22 + a_6_65·b_1_22 + a_2_8·b_1_26
       + a_2_8·b_1_1·b_1_25 + a_2_8·b_2_9·b_1_2·b_3_19 + a_2_8·b_2_93
  20. a_2_8·a_6_65
  21. b_1_24·a_5_47 + a_6_65·b_3_19 + b_2_9·b_1_22·a_5_47 + b_2_9·a_6_65·a_1_0
  22. b_2_94·a_1_0·b_1_2 + a_2_8·b_2_92·b_1_2·b_3_19 + a_2_8·b_2_94 + a_5_472
       + b_2_92·a_1_0·a_5_47
  23. a_5_47·b_5_52 + b_2_92·a_6_65 + a_2_8·b_2_92·b_1_2·b_3_19 + b_2_92·a_1_0·a_5_47
  24. b_5_522 + b_1_1·b_1_24·b_5_52 + b_1_15·b_5_52 + b_2_93·b_1_24 + b_2_94·b_1_22
       + b_2_94·a_1_0·b_1_2 + a_2_8·b_1_28 + a_2_8·b_1_1·b_1_27
       + a_2_8·b_1_12·b_1_2·b_5_52 + a_2_8·b_1_14·b_1_24 + a_2_8·b_1_15·b_1_23
       + c_8_146·b_1_12
  25. a_6_65·a_5_47 + b_2_92·a_6_65·a_1_0
  26. a_6_65·b_5_52 + b_2_9·a_6_65·b_3_19 + a_2_8·b_1_1·b_1_23·b_5_52
       + a_2_8·b_1_14·b_5_52 + b_2_92·a_6_65·a_1_0 + a_2_8·c_8_146·b_1_1
  27. a_6_652

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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_1_3, a Duflot regular element of degree 1
    2. c_8_146, a Duflot regular element of degree 8
    3. b_1_24 + b_1_12·b_1_22 + b_1_14 + b_2_9·b_1_22 + b_2_92, an element of degree 4
    4. b_3_19 + b_1_1·b_1_22 + b_1_12·b_1_2, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 9, 12].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.

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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. c_1_3 → c_1_0, an element of degree 1
  5. a_2_8 → 0, an element of degree 2
  6. b_2_9 → 0, an element of degree 2
  7. b_3_19 → 0, an element of degree 3
  8. a_5_47 → 0, an element of degree 5
  9. b_5_52 → 0, an element of degree 5
  10. a_6_65 → 0, an element of degree 6
  11. c_8_146 → c_1_18, an element of degree 8

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. c_1_3 → c_1_0, an element of degree 1
  5. a_2_8 → 0, an element of degree 2
  6. b_2_9 → 0, an element of degree 2
  7. b_3_19 → 0, an element of degree 3
  8. a_5_47 → 0, an element of degree 5
  9. b_5_52 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  10. a_6_65 → 0, an element of degree 6
  11. c_8_146 → c_1_12·c_1_22·c_1_34 + c_1_12·c_1_26 + c_1_14·c_1_34 + c_1_18, an element of degree 8

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. c_1_3 → c_1_0, an element of degree 1
  5. a_2_8 → 0, an element of degree 2
  6. b_2_9 → c_1_32, an element of degree 2
  7. b_3_19 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  8. a_5_47 → 0, an element of degree 5
  9. b_5_52 → c_1_2·c_1_34 + c_1_22·c_1_33, an element of degree 5
  10. a_6_65 → 0, an element of degree 6
  11. c_8_146 → c_1_38 + c_1_22·c_1_36 + c_1_24·c_1_34 + c_1_27·c_1_3
       + c_1_12·c_1_22·c_1_34 + c_1_12·c_1_24·c_1_32 + c_1_14·c_1_34
       + c_1_14·c_1_22·c_1_32 + c_1_14·c_1_24 + c_1_18, an element of degree 8

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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