Cohomology of group number 600 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 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
     − 1
    (t  −  1)3 · (t4  +  1)
  • 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 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. 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_3_9, an element of degree 3
  8. b_5_17, an element of degree 5
  9. a_6_14, a nilpotent element of degree 6
  10. a_7_11, a nilpotent element of degree 7
  11. c_8_30, a Duflot regular element of degree 8

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Ring relations

There are 30 minimal relations of maximal degree 14:

  1. a_1_12 + a_1_02
  2. a_1_0·a_1_1
  3. a_1_0·b_1_2
  4. b_2_3·a_1_0
  5. b_2_4·b_1_2 + b_2_3·b_1_2 + b_2_5·a_1_1 + b_2_3·a_1_1
  6. b_2_3·b_1_22 + b_2_32 + b_2_5·a_1_1·b_1_2 + b_2_5·a_1_02
  7. b_1_2·b_3_9 + b_2_3·b_2_5 + b_2_32 + b_2_5·a_1_1·b_1_2 + b_2_3·a_1_1·b_1_2
  8. b_2_3·b_1_22 + b_2_3·b_2_4 + a_1_1·b_3_9 + b_2_5·a_1_1·b_1_2
  9. b_2_52·a_1_1 + b_2_4·b_2_5·a_1_1 + b_2_3·b_2_5·a_1_1 + a_1_02·b_3_9
  10. b_2_3·b_3_9 + b_2_3·b_2_5·b_1_2 + b_2_32·b_1_2 + b_2_52·a_1_1 + b_2_4·b_2_5·a_1_1
       + b_2_3·b_2_5·a_1_1 + b_2_32·a_1_1
  11. b_1_2·b_5_17 + b_2_3·b_2_52 + b_2_33 + b_2_3·b_2_5·a_1_1·b_1_2
  12. a_1_1·b_5_17 + b_2_4·a_1_1·b_3_9 + b_2_3·b_2_5·a_1_1·b_1_2 + b_2_4·b_2_5·a_1_02
  13. b_3_92 + b_2_4·b_2_52 + b_2_42·b_2_5 + b_2_32·b_2_5 + b_2_33 + a_1_0·b_5_17
       + b_2_5·a_1_1·b_3_9
  14. b_2_3·b_5_17 + b_2_3·b_2_52·b_1_2 + b_2_33·b_1_2 + b_2_5·a_1_02·b_3_9
       + b_2_4·a_1_02·b_3_9
  15. a_6_14·a_1_1 + b_2_5·a_1_02·b_3_9 + b_2_4·a_1_02·b_3_9
  16. a_6_14·a_1_0
  17. b_2_5·a_1_0·b_5_17 + b_2_52·a_1_0·b_3_9 + b_2_4·a_6_14 + b_2_4·b_2_5·a_1_1·b_3_9
       + b_2_42·a_1_1·b_3_9 + b_2_3·a_6_14 + b_2_33·a_1_1·b_1_2
  18. b_1_2·a_7_11 + a_6_14·b_1_22 + b_2_3·a_6_14 + b_2_33·a_1_1·b_1_2
  19. a_1_1·a_7_11
  20. a_1_0·a_7_11 + b_2_42·b_2_5·a_1_02
  21. b_2_3·a_7_11 + b_2_34·a_1_1
  22. a_6_14·b_3_9 + b_2_5·a_7_11 + b_2_5·a_6_14·b_1_2 + b_2_54·a_1_0 + b_2_4·b_2_53·a_1_0
       + b_2_42·b_2_52·a_1_0 + b_2_3·a_6_14·b_1_2 + b_2_33·b_2_5·a_1_1
       + b_2_42·a_1_02·b_3_9
  23. a_1_0·b_3_9·b_5_17 + b_2_4·a_7_11 + b_2_43·b_2_5·a_1_0 + b_2_34·a_1_1
  24. b_3_9·a_7_11 + b_2_53·a_1_0·b_3_9 + b_2_4·b_2_5·a_6_14 + b_2_4·b_2_52·a_1_0·b_3_9
       + b_2_42·a_6_14 + b_2_42·b_2_5·a_1_1·b_3_9 + b_2_42·b_2_5·a_1_0·b_3_9
       + b_2_43·a_1_1·b_3_9 + b_2_3·b_2_5·a_6_14 + b_2_32·a_6_14 + b_2_33·b_2_5·a_1_1·b_1_2
  25. b_5_172 + b_2_4·b_2_54 + b_2_44·b_2_5 + b_2_34·b_2_5 + b_2_35
       + b_2_53·a_1_0·b_3_9 + b_2_4·b_2_5·a_6_14 + b_2_42·a_1_0·b_5_17 + b_2_42·a_6_14
       + b_2_42·b_2_5·a_1_0·b_3_9 + b_2_43·a_1_1·b_3_9 + b_2_3·b_2_5·a_6_14 + b_2_32·a_6_14
       + b_2_34·a_1_1·b_1_2 + c_8_30·a_1_02
  26. a_6_14·b_5_17 + b_2_52·a_7_11 + b_2_52·a_6_14·b_1_2 + b_2_55·a_1_0
       + b_2_42·b_2_53·a_1_0 + b_2_43·b_2_52·a_1_0 + b_2_32·a_6_14·b_1_2
       + b_2_43·a_1_02·b_3_9
  27. a_6_142
  28. b_5_17·a_7_11 + b_2_54·a_1_0·b_3_9 + b_2_42·b_2_52·a_1_0·b_3_9 + b_2_43·a_6_14
       + b_2_43·b_2_5·a_1_1·b_3_9 + b_2_43·b_2_5·a_1_0·b_3_9 + b_2_44·a_1_1·b_3_9
       + b_2_33·a_6_14
  29. a_6_14·a_7_11
  30. a_7_112 + b_2_45·b_2_5·a_1_02

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 14.
  • 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_30, a Duflot regular element of degree 8
    2. b_1_24 + b_2_52 + b_2_4·b_2_5 + b_2_42 + b_2_3·b_2_5 + b_2_32, an element of degree 4
    3. b_2_5, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 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. 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_3_9 → 0, an element of degree 3
  8. b_5_17 → 0, an element of degree 5
  9. a_6_14 → 0, an element of degree 6
  10. a_7_11 → 0, an element of degree 7
  11. c_8_30 → 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. b_1_2 → c_1_1, 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 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  7. b_3_9 → 0, an element of degree 3
  8. b_5_17 → 0, an element of degree 5
  9. a_6_14 → 0, an element of degree 6
  10. a_7_11 → 0, an element of degree 7
  11. c_8_30 → 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

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. 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_12, an element of degree 2
  6. b_2_5 → c_1_22, an element of degree 2
  7. b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_5_17 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  9. a_6_14 → 0, an element of degree 6
  10. a_7_11 → 0, an element of degree 7
  11. c_8_30 → 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

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. b_1_2 → c_1_2, an element of degree 1
  4. b_2_3 → c_1_22, an element of degree 2
  5. b_2_4 → c_1_22, an element of degree 2
  6. b_2_5 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  7. b_3_9 → c_1_23 + c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_5_17 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  9. a_6_14 → 0, an element of degree 6
  10. a_7_11 → 0, an element of degree 7
  11. c_8_30 → c_1_12·c_1_26 + c_1_14·c_1_24 + 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