Cohomology of group number 590 of order 128

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

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

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring generators

The cohomology ring has 13 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. a_4_9, a nilpotent element of degree 4
  8. b_5_15, an element of degree 5
  9. a_6_15, a nilpotent element of degree 6
  10. a_7_20, a nilpotent element of degree 7
  11. b_7_22, an element of degree 7
  12. b_8_27, an element of degree 8
  13. c_8_28, a Duflot regular element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring relations

There are 50 minimal relations of maximal degree 16:

  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_4·a_1_1 + b_2_4·a_1_0 + 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_3·a_1_1·b_1_2
  7. b_2_52·a_1_1 + b_2_4·b_2_5·a_1_1 + b_2_3·b_2_5·a_1_1
  8. a_4_9·b_1_2 + b_2_3·b_2_4·a_1_1
  9. a_4_9·a_1_1
  10. b_2_3·b_2_4·a_1_1 + b_2_32·a_1_1 + a_4_9·a_1_0
  11. b_2_3·a_4_9 + b_2_32·a_1_1·b_1_2
  12. b_1_2·b_5_15 + b_2_3·b_2_52 + b_2_3·b_2_4·b_2_5 + b_2_32·b_2_5 + b_2_33
  13. a_1_1·b_5_15 + b_2_32·a_1_1·b_1_2
  14. b_2_3·b_5_15 + b_2_3·b_2_52·b_1_2 + b_2_33·b_1_2 + b_2_4·b_2_52·a_1_0
       + b_2_42·b_2_5·a_1_0 + b_2_32·b_2_5·a_1_1 + b_2_33·a_1_1 + b_2_5·a_4_9·a_1_0
       + b_2_4·a_4_9·a_1_0
  15. a_6_15·a_1_1 + b_2_5·a_4_9·a_1_0 + b_2_4·a_4_9·a_1_0
  16. a_6_15·a_1_0 + b_2_5·a_4_9·a_1_0
  17. a_4_92
  18. b_2_3·b_2_42·b_2_5 + b_2_3·b_2_43 + b_2_33·b_2_5 + b_2_34 + b_2_5·a_1_0·b_5_15
       + b_2_52·a_4_9 + b_2_4·a_6_15 + b_2_3·a_6_15 + b_2_32·b_2_5·a_1_1·b_1_2
  19. b_1_2·a_7_20 + a_6_15·b_1_22 + b_2_3·a_6_15 + b_2_33·a_1_1·b_1_2
  20. a_1_1·a_7_20
  21. a_1_0·a_7_20
  22. b_1_2·b_7_22 + b_2_33·b_2_5 + b_2_5·a_1_0·b_5_15 + b_2_52·a_4_9 + b_2_4·b_2_5·a_4_9
       + b_2_3·a_6_15
  23. a_1_1·b_7_22 + b_2_32·b_2_5·a_1_1·b_1_2
  24. a_1_0·b_7_22 + b_2_5·a_1_0·b_5_15 + b_2_52·a_4_9 + b_2_4·b_2_5·a_4_9
       + b_2_32·b_2_5·a_1_1·b_1_2
  25. b_2_3·a_7_20 + b_2_34·a_1_1 + b_2_4·b_2_5·a_4_9·a_1_0 + b_2_42·a_4_9·a_1_0
  26. a_4_9·b_5_15 + b_2_4·a_7_20 + b_2_42·b_2_52·a_1_0 + b_2_43·b_2_5·a_1_1
       + b_2_44·a_1_1 + b_2_44·a_1_0 + b_2_33·b_2_5·a_1_1 + b_2_34·a_1_1
  27. b_2_3·b_7_22 + b_2_33·b_2_5·b_1_2 + a_4_9·b_5_15 + b_2_4·b_2_53·a_1_0
       + b_2_42·b_2_52·a_1_0 + b_2_3·a_6_15·b_1_2 + b_2_34·a_1_1 + b_2_4·b_2_5·a_4_9·a_1_0
  28. b_8_27·b_1_2 + b_2_54·b_1_2 + b_2_34·b_1_2 + a_4_9·b_5_15 + b_2_4·b_2_53·a_1_0
       + b_2_42·b_2_52·a_1_0 + b_2_43·b_2_5·a_1_0 + b_2_3·a_6_15·b_1_2
       + b_2_33·b_2_5·a_1_1 + b_2_4·b_2_5·a_4_9·a_1_0 + b_2_42·a_4_9·a_1_0
  29. b_8_27·a_1_1 + b_2_34·a_1_1 + b_2_42·a_4_9·a_1_0
  30. b_8_27·a_1_0 + a_4_9·b_5_15 + 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_43·b_2_5·a_1_0 + b_2_34·a_1_1
  31. a_4_9·a_6_15
  32. b_5_152 + b_2_4·b_2_54 + b_2_43·b_2_52 + b_2_33·b_2_52 + b_2_35
       + b_2_53·a_4_9 + b_2_4·b_2_5·a_1_0·b_5_15 + b_2_4·b_2_5·a_6_15 + b_2_42·a_1_0·b_5_15
       + b_2_3·b_2_5·a_6_15 + b_2_33·b_2_5·a_1_1·b_1_2 + b_2_34·a_1_1·b_1_2
       + c_8_28·a_1_02
  33. b_2_3·b_8_27 + b_2_3·b_2_54 + b_2_3·b_2_44 + b_2_42·a_6_15 + b_2_42·b_2_5·a_4_9
       + b_2_34·a_1_1·b_1_2
  34. a_4_9·a_7_20 + b_2_42·b_2_5·a_4_9·a_1_0 + b_2_43·a_4_9·a_1_0
  35. a_6_15·b_5_15 + b_2_52·a_7_20 + b_2_52·a_6_15·b_1_2 + b_2_55·a_1_0
       + b_2_4·b_2_54·a_1_0 + b_2_43·b_2_52·a_1_0 + b_2_44·b_2_5·a_1_0
       + b_2_32·a_6_15·b_1_2 + b_2_42·b_2_5·a_4_9·a_1_0 + b_2_43·a_4_9·a_1_0
  36. a_4_9·b_7_22 + b_2_4·b_2_5·a_7_20 + b_2_4·b_2_54·a_1_0 + b_2_42·b_2_53·a_1_0
       + b_2_43·b_2_52·a_1_0 + b_2_44·b_2_5·a_1_0 + b_2_43·a_4_9·a_1_0
  37. a_6_152
  38. b_5_15·a_7_20 + b_2_54·a_4_9 + b_2_4·b_2_53·a_4_9 + b_2_42·b_2_5·a_6_15
       + b_2_42·b_2_52·a_4_9 + b_2_32·b_2_5·a_6_15 + b_2_35·a_1_1·b_1_2
  39. a_4_9·b_8_27 + b_2_54·a_4_9 + b_2_42·b_2_5·a_1_0·b_5_15 + b_2_42·b_2_5·a_6_15
       + b_2_42·b_2_52·a_4_9 + b_2_43·a_1_0·b_5_15 + b_2_43·b_2_5·a_4_9
       + b_2_32·b_2_5·a_6_15 + b_2_34·b_2_5·a_1_1·b_1_2 + b_2_35·a_1_1·b_1_2
  40. b_5_15·b_7_22 + b_2_52·b_8_27 + b_2_56 + b_2_4·b_2_5·b_8_27 + b_2_4·b_2_55
       + b_2_43·b_2_53 + b_2_44·b_2_52 + b_2_4·b_2_52·a_6_15 + b_2_4·b_2_53·a_4_9
       + b_2_42·b_2_5·a_1_0·b_5_15 + b_2_3·b_2_52·a_6_15 + b_2_32·b_2_5·a_6_15
       + b_2_33·a_6_15 + b_2_34·b_2_5·a_1_1·b_1_2 + b_2_5·c_8_28·a_1_02
  41. a_6_15·a_7_20 + b_2_43·b_2_5·a_4_9·a_1_0 + b_2_44·a_4_9·a_1_0
  42. a_6_15·b_7_22 + b_2_4·b_2_52·a_7_20 + b_2_42·b_2_5·a_7_20 + b_2_43·a_7_20
       + b_2_43·b_2_53·a_1_0 + b_2_44·b_2_52·a_1_0 + b_2_45·b_2_5·a_1_1
       + b_2_45·b_2_5·a_1_0 + b_2_46·a_1_1 + b_2_46·a_1_0 + b_2_32·b_2_5·a_6_15·b_1_2
  43. b_8_27·b_5_15 + b_2_54·b_5_15 + b_2_4·b_2_52·b_7_22 + b_2_42·b_2_5·b_7_22
       + b_2_43·b_2_5·b_5_15 + b_2_35·b_2_5·b_1_2 + b_2_36·b_1_2 + b_2_42·b_2_5·a_7_20
       + b_2_42·b_2_54·a_1_0 + b_2_45·b_2_5·a_1_0 + b_2_32·b_2_5·a_6_15·b_1_2
       + b_2_33·a_6_15·b_1_2 + b_2_43·b_2_5·a_4_9·a_1_0 + a_4_9·c_8_28·a_1_0
  44. a_7_202
  45. a_7_20·b_7_22 + b_2_4·b_2_53·a_6_15 + b_2_42·b_2_52·a_6_15 + b_2_43·b_2_52·a_4_9
       + b_2_44·b_2_5·a_4_9 + b_2_3·b_2_53·a_6_15 + b_2_32·b_2_52·a_6_15
  46. b_7_222 + b_2_42·b_2_55 + b_2_44·b_2_53 + b_2_32·b_2_55 + b_2_34·b_2_53
       + b_2_35·b_2_52
  47. a_6_15·b_8_27 + b_2_54·a_6_15 + b_2_4·b_2_54·a_4_9 + b_2_42·b_2_53·a_4_9
       + b_2_43·b_2_5·a_1_0·b_5_15 + b_2_44·a_1_0·b_5_15 + b_2_44·b_2_5·a_4_9
       + b_2_34·a_6_15 + b_2_35·b_2_5·a_1_1·b_1_2
  48. b_8_27·a_7_20 + b_2_54·a_7_20 + b_2_4·b_2_53·a_7_20 + b_2_4·b_2_56·a_1_0
       + b_2_42·b_2_55·a_1_0 + b_2_43·b_2_5·a_7_20 + b_2_44·a_7_20 + b_2_44·b_2_53·a_1_0
       + b_2_46·b_2_5·a_1_1 + b_2_47·a_1_1 + b_2_47·a_1_0 + b_2_37·a_1_1
       + b_2_45·a_4_9·a_1_0
  49. b_8_27·b_7_22 + b_2_54·b_7_22 + b_2_42·b_2_53·b_5_15 + b_2_43·b_2_5·b_7_22
       + b_2_43·b_2_52·b_5_15 + b_2_32·b_2_55·b_1_2 + b_2_33·b_2_54·b_1_2
       + b_2_34·b_2_53·b_1_2 + b_2_36·b_2_5·b_1_2 + b_2_42·b_2_55·a_1_0
       + b_2_43·b_2_54·a_1_0 + b_2_34·a_6_15·b_1_2 + b_2_36·b_2_5·a_1_1
       + b_2_44·b_2_5·a_4_9·a_1_0 + b_2_5·a_4_9·c_8_28·a_1_0
  50. b_8_272 + b_2_58 + b_2_43·b_2_55 + b_2_45·b_2_53 + b_2_46·b_2_52
       + b_2_33·b_2_55 + b_2_35·b_2_53 + b_2_36·b_2_52 + b_2_38
       + b_2_42·b_2_54·a_4_9 + b_2_43·b_2_52·a_6_15 + b_2_43·b_2_53·a_4_9
       + b_2_44·b_2_5·a_1_0·b_5_15 + b_2_44·b_2_5·a_6_15 + b_2_45·a_1_0·b_5_15
       + b_2_33·b_2_52·a_6_15 + b_2_34·b_2_5·a_6_15

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

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_28, 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_3·b_2_4, an element of degree 4
    3. b_2_5, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

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. a_4_9 → 0, an element of degree 4
  8. b_5_15 → 0, an element of degree 5
  9. a_6_15 → 0, an element of degree 6
  10. a_7_20 → 0, an element of degree 7
  11. b_7_22 → 0, an element of degree 7
  12. b_8_27 → 0, an element of degree 8
  13. c_8_28 → 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. a_4_9 → 0, an element of degree 4
  8. b_5_15 → 0, an element of degree 5
  9. a_6_15 → 0, an element of degree 6
  10. a_7_20 → 0, an element of degree 7
  11. b_7_22 → 0, an element of degree 7
  12. b_8_27 → c_1_28 + c_1_14·c_1_24, an element of degree 8
  13. c_8_28 → 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. a_4_9 → 0, an element of degree 4
  8. b_5_15 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  9. a_6_15 → 0, an element of degree 6
  10. a_7_20 → 0, an element of degree 7
  11. b_7_22 → c_1_1·c_1_26 + c_1_12·c_1_25, an element of degree 7
  12. b_8_27 → c_1_28 + c_1_14·c_1_24 + c_1_18, an element of degree 8
  13. c_8_28 → 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_22, an element of degree 2
  6. b_2_5 → c_1_12, an element of degree 2
  7. a_4_9 → 0, an element of degree 4
  8. b_5_15 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  9. a_6_15 → 0, an element of degree 6
  10. a_7_20 → 0, an element of degree 7
  11. b_7_22 → c_1_13·c_1_24 + c_1_15·c_1_22, an element of degree 7
  12. b_8_27 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_18, an element of degree 8
  13. c_8_28 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
       + 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

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