Cohomology of group number 633 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 a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t5  −  3·t4  +  3·t3  −  2·t2  +  t  −  1)
    (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-4,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 14 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. a_2_3, a nilpotent element of degree 2
  5. b_2_4, an element of degree 2
  6. b_2_5, an element of degree 2
  7. c_2_6, a Duflot regular element of degree 2
  8. a_3_11, a nilpotent element of degree 3
  9. a_5_22, a nilpotent element of degree 5
  10. b_5_23, an element of degree 5
  11. b_5_25, an element of degree 5
  12. a_6_34, a nilpotent element of degree 6
  13. b_6_32, an element of degree 6
  14. c_8_66, a Duflot regular element of degree 8

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Ring relations

There are 53 minimal relations of maximal degree 12:

  1. a_1_12 + a_1_02
  2. a_1_0·a_1_1 + a_1_02
  3. a_1_0·b_1_2
  4. a_2_3·a_1_1 + a_2_3·a_1_0
  5. b_2_4·a_1_1 + a_2_3·b_1_2
  6. b_2_4·a_1_0
  7. b_2_5·a_1_1 + b_2_5·a_1_0 + a_1_03
  8. a_2_32 + a_2_3·a_1_02
  9. a_2_3·b_2_4
  10. b_2_5·b_1_22 + b_2_42 + a_2_3·b_1_22
  11. a_2_3·b_2_5 + a_1_1·a_3_11 + a_2_3·a_1_02
  12. a_2_3·b_2_5 + a_1_0·a_3_11
  13. b_2_5·a_1_03
  14. a_2_3·a_3_11 + a_1_02·a_3_11
  15. a_3_112 + b_2_5·a_1_0·a_3_11 + b_2_52·a_1_02 + a_1_03·a_3_11
  16. a_1_1·a_5_22 + b_2_52·a_1_02
  17. a_1_0·a_5_22 + b_2_52·a_1_02 + a_1_03·a_3_11
  18. b_1_2·a_5_22 + a_1_1·b_5_23
  19. a_1_0·b_5_23
  20. b_1_2·b_5_25 + b_2_4·b_2_52 + b_1_2·a_5_22 + b_2_5·b_1_2·a_3_11
  21. a_1_1·b_5_25 + a_1_0·b_5_25
  22. a_2_3·a_5_22 + b_2_5·a_1_02·a_3_11
  23. b_2_4·a_5_22 + a_2_3·b_5_23
  24. b_2_5·a_5_22 + b_2_53·a_1_0 + a_1_02·b_5_25
  25. b_2_53·b_1_2 + b_2_4·b_5_25 + b_2_4·a_5_22 + b_2_4·b_2_5·a_3_11
  26. a_6_34·b_1_2 + b_2_4·a_5_22 + b_2_4·b_2_5·a_3_11
  27. b_2_5·a_5_22 + b_2_53·a_1_0 + a_2_3·b_5_25 + a_6_34·a_1_1 + b_2_5·a_1_02·a_3_11
  28. b_2_5·a_5_22 + b_2_53·a_1_0 + a_2_3·b_5_25 + a_6_34·a_1_0 + b_2_5·a_1_02·a_3_11
  29. b_6_32·b_1_2 + b_2_4·b_5_23 + b_2_4·b_2_52·b_1_2
  30. b_6_32·a_1_1 + b_2_4·a_5_22
  31. b_6_32·a_1_0
  32. a_3_11·a_5_22 + b_2_52·a_1_0·a_3_11 + a_6_34·a_1_02
  33. a_3_11·b_5_25 + b_2_5·a_1_0·b_5_25 + b_2_5·a_6_34 + a_3_11·a_5_22 + b_2_53·a_1_02
  34. b_2_52·b_1_2·a_3_11 + b_2_4·a_6_34
  35. a_2_3·a_6_34
  36. b_2_5·b_1_2·b_5_23 + b_2_4·b_6_32 + b_2_42·b_2_52 + a_2_3·b_1_2·b_5_23
       + c_2_6·a_1_03·a_3_11
  37. a_2_3·b_6_32
  38. a_6_34·a_3_11 + b_2_5·a_1_02·b_5_25
  39. a_5_222 + b_2_54·a_1_02
  40. a_5_22·b_5_25 + b_2_52·a_1_0·b_5_25 + b_2_54·a_1_02
  41. b_5_23·b_5_25 + b_2_52·b_6_32 + b_2_4·b_2_54 + a_5_22·b_5_23 + b_2_5·a_3_11·b_5_23
  42. b_5_232 + c_8_66·b_1_22
  43. a_5_22·b_5_23 + c_8_66·a_1_1·b_1_2
  44. b_5_252 + b_2_55 + a_5_22·b_5_25 + b_2_54·a_1_02 + c_8_66·a_1_02
  45. a_6_34·a_5_22 + b_2_52·a_6_34·a_1_0 + b_2_53·a_1_02·a_3_11
  46. b_6_32·b_5_25 + b_2_53·b_5_23 + b_2_4·b_2_52·b_5_25 + a_6_34·b_5_23
  47. b_6_32·a_5_22 + a_6_34·b_5_23 + b_2_5·b_6_32·a_3_11 + b_2_4·b_2_53·a_3_11
  48. b_6_32·b_5_23 + b_2_4·b_2_52·b_5_23 + b_2_4·c_8_66·b_1_2
  49. b_6_32·a_5_22 + a_2_3·c_8_66·b_1_2
  50. a_6_34·b_5_25 + b_2_54·a_3_11 + b_2_55·a_1_0 + b_2_53·a_1_02·a_3_11
       + a_2_3·c_8_66·a_1_0 + c_8_66·a_1_03
  51. a_6_34·b_6_32 + b_2_52·a_3_11·b_5_23 + b_2_4·b_2_52·a_6_34
  52. b_6_322 + b_2_42·b_2_54 + b_2_42·c_8_66
  53. a_6_342 + b_2_54·a_1_0·a_3_11 + b_2_52·a_6_34·a_1_02 + a_2_3·c_8_66·a_1_02

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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_2_6, a Duflot regular element of degree 2
    2. c_8_66, a Duflot regular element of degree 8
    3. b_1_22 + b_2_5 + b_2_4, 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, 6, 8, 10].
  • 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. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_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. c_2_6 → c_1_02, an element of degree 2
  8. a_3_11 → 0, an element of degree 3
  9. a_5_22 → 0, an element of degree 5
  10. b_5_23 → 0, an element of degree 5
  11. b_5_25 → 0, an element of degree 5
  12. a_6_34 → 0, an element of degree 6
  13. b_6_32 → 0, an element of degree 6
  14. c_8_66 → 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. a_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_2·c_1_3, an element of degree 2
  6. b_2_5 → c_1_32, an element of degree 2
  7. c_2_6 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  8. a_3_11 → 0, an element of degree 3
  9. a_5_22 → 0, an element of degree 5
  10. b_5_23 → c_1_2·c_1_34 + c_1_12·c_1_2·c_1_32 + c_1_14·c_1_2, an element of degree 5
  11. b_5_25 → c_1_35, an element of degree 5
  12. a_6_34 → 0, an element of degree 6
  13. b_6_32 → c_1_12·c_1_2·c_1_33 + c_1_14·c_1_2·c_1_3, an element of degree 6
  14. c_8_66 → c_1_38 + c_1_14·c_1_34 + c_1_18, an element of degree 8

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