Cohomology of group number 565 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 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t6  −  t5  −  t4  +  t3  −  1
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-5,-3,-3. 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. 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. b_2_4, an element of degree 2
  6. a_4_3, a nilpotent element of degree 4
  7. b_5_7, an element of degree 5
  8. b_5_8, an element of degree 5
  9. b_5_9, an element of degree 5
  10. b_8_14, an element of degree 8
  11. c_8_17, a Duflot regular element of degree 8

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Ring relations

There are 30 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_12 + a_1_0·a_1_2
  3. a_1_22 + a_1_0·a_1_1
  4. b_2_4·a_1_0 + b_2_3·a_1_2 + b_2_3·a_1_1
  5. b_2_3·b_2_4·a_1_2 + b_2_3·b_2_4·a_1_1 + b_2_32·a_1_2 + b_2_32·a_1_1
  6. a_4_3·a_1_0
  7. a_4_3·a_1_2 + a_4_3·a_1_1
  8. a_1_0·b_5_7 + b_2_3·a_4_3 + b_2_32·a_1_1·a_1_2 + b_2_32·a_1_0·a_1_1
  9. a_1_2·b_5_7 + a_1_1·b_5_7 + a_1_0·b_5_8
  10. a_1_2·b_5_8 + a_1_2·b_5_7 + a_1_1·b_5_8 + a_1_1·b_5_7
  11. a_1_0·b_5_9 + b_2_3·a_4_3 + b_2_32·a_1_1·a_1_2
  12. a_1_2·b_5_9 + a_1_1·b_5_9 + b_2_4·a_4_3
  13. b_2_4·b_5_7 + b_2_3·b_5_8 + b_2_33·a_1_2 + b_2_33·a_1_1 + b_2_4·a_4_3·a_1_1
  14. a_4_32
  15. a_4_3·b_5_7 + b_2_34·a_1_2 + b_2_34·a_1_1 + b_2_3·b_2_4·a_4_3·a_1_1
  16. a_4_3·b_5_8 + b_2_34·a_1_2 + b_2_34·a_1_1 + b_2_3·b_2_4·a_4_3·a_1_1
  17. a_4_3·b_5_9 + b_2_34·a_1_2 + b_2_34·a_1_1 + b_2_3·b_2_4·a_4_3·a_1_1
       + b_2_32·a_4_3·a_1_1
  18. b_8_14·a_1_0
  19. b_8_14·a_1_2 + b_8_14·a_1_1
  20. b_5_72 + b_2_34·b_2_4 + b_2_33·a_4_3 + b_2_34·a_1_1·a_1_2 + b_2_34·a_1_0·a_1_1
  21. b_5_82 + b_5_7·b_5_8 + b_2_32·b_2_43 + b_2_33·b_2_42 + b_2_32·b_2_4·a_4_3
  22. b_5_92 + b_2_3·b_2_44 + b_2_33·b_2_42 + b_2_34·b_2_4 + b_2_33·a_4_3
       + b_2_34·a_1_1·a_1_2
  23. b_5_82 + b_2_32·b_2_43 + b_2_32·b_2_4·a_4_3 + b_2_34·a_1_1·a_1_2
       + b_2_34·a_1_0·a_1_1 + c_8_17·a_1_0·a_1_2 + c_8_17·a_1_0·a_1_1
  24. b_5_7·b_5_9 + b_2_3·b_8_14 + b_2_32·b_2_43 + b_2_32·a_1_1·b_5_9
       + b_2_32·b_2_4·a_4_3 + b_2_34·a_1_0·a_1_1
  25. b_5_8·b_5_9 + b_2_4·b_8_14 + b_2_3·b_2_44 + b_2_43·a_4_3 + b_2_3·b_2_4·a_1_1·b_5_9
       + b_2_32·b_2_4·a_4_3
  26. a_4_3·b_8_14
  27. b_8_14·b_5_8 + b_2_3·b_2_43·b_5_8 + b_2_32·b_2_42·b_5_9 + b_2_3·b_2_4·b_8_14·a_1_1
       + b_2_32·b_2_44·a_1_1 + b_2_36·a_1_2 + b_2_36·a_1_1
  28. b_8_14·b_5_9 + b_2_3·b_2_43·b_5_9 + b_2_3·b_2_43·b_5_8 + b_2_33·b_2_4·b_5_8
       + b_2_34·b_5_8 + b_2_32·b_2_44·a_1_1 + b_2_34·b_2_42·a_1_1 + b_2_35·b_2_4·a_1_1
       + b_2_36·a_1_2 + b_2_36·a_1_1 + b_2_33·b_2_4·a_4_3·a_1_1 + b_2_34·a_4_3·a_1_1
  29. b_8_14·b_5_7 + b_2_32·b_2_42·b_5_8 + b_2_33·b_2_4·b_5_9 + b_2_32·b_8_14·a_1_1
       + b_2_33·b_2_43·a_1_1 + b_2_36·a_1_2 + b_2_36·a_1_1 + b_2_33·b_2_4·a_4_3·a_1_1
       + b_2_34·a_4_3·a_1_1
  30. b_8_142 + b_2_32·b_2_46 + b_2_33·b_2_45 + b_2_35·b_2_43 + b_2_36·b_2_42

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 16.
  • 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_17, a Duflot regular element of degree 8
    2. b_2_42 + b_2_3·b_2_4 + b_2_32, an element of degree 4
    3. b_2_4, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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. a_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. a_4_3 → 0, an element of degree 4
  7. b_5_7 → 0, an element of degree 5
  8. b_5_8 → 0, an element of degree 5
  9. b_5_9 → 0, an element of degree 5
  10. b_8_14 → 0, an element of degree 8
  11. c_8_17 → 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. a_1_2 → 0, an element of degree 1
  4. b_2_3 → c_1_22, an element of degree 2
  5. b_2_4 → c_1_12, an element of degree 2
  6. a_4_3 → 0, an element of degree 4
  7. b_5_7 → c_1_1·c_1_24, an element of degree 5
  8. b_5_8 → c_1_13·c_1_22, an element of degree 5
  9. b_5_9 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  10. b_8_14 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22, an element of degree 8
  11. c_8_17 → 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

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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