Cohomology of group number 558 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 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
    (t  +  1) · (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-4,-4,-4. 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 14 minimal generators of maximal degree 5:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_1, 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. a_3_4, a nilpotent element of degree 3
  8. a_3_7, a nilpotent element of degree 3
  9. a_3_9, a nilpotent element of degree 3
  10. a_4_10, a nilpotent element of degree 4
  11. b_4_13, an element of degree 4
  12. c_4_14, a Duflot regular element of degree 4
  13. c_4_15, a Duflot regular element of degree 4
  14. a_5_18, a nilpotent element of degree 5

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

Ring relations

There are 53 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·a_1_2
  3. a_1_0·b_1_1 + a_1_22
  4. a_1_22·b_1_1
  5. a_2_3·a_1_0
  6. b_2_4·a_1_0 + a_2_3·a_1_2
  7. b_2_4·a_1_2 + a_2_3·b_1_1
  8. b_2_5·a_1_2 + b_2_5·a_1_0
  9. a_2_32
  10. a_2_3·b_2_4
  11. b_2_5·b_1_12 + b_2_42 + a_2_3·a_1_2·b_1_1
  12. a_1_0·a_3_4 + a_2_3·a_1_2·b_1_1
  13. a_1_2·a_3_4
  14. a_1_0·a_3_7
  15. b_1_1·a_3_4 + a_2_3·b_1_12 + a_1_2·a_3_7
  16. a_2_3·b_2_5 + a_1_0·a_3_9
  17. a_2_3·b_2_5 + a_1_2·a_3_9
  18. a_2_3·a_3_4
  19. b_2_4·a_3_4 + a_2_3·a_3_7
  20. b_1_12·a_3_9 + b_2_5·a_3_7 + b_2_52·a_1_0
  21. a_2_3·a_3_9
  22. a_4_10·b_1_1 + b_2_5·a_3_7 + b_2_52·a_1_0 + b_2_4·a_3_7 + b_2_4·a_3_4 + a_2_3·b_1_13
  23. a_4_10·a_1_0
  24. b_2_4·a_3_4 + a_4_10·a_1_2
  25. b_4_13·a_1_0 + b_2_5·a_3_4 + b_2_52·a_1_0
  26. b_4_13·a_1_2 + b_2_5·a_3_4 + b_2_52·a_1_0
  27. a_3_42
  28. a_3_4·a_3_7 + a_2_3·b_1_1·a_3_7
  29. a_3_92
  30. a_3_7·a_3_9 + b_2_5·a_1_0·a_3_9
  31. a_2_3·b_1_14 + a_3_72 + a_1_2·b_1_12·a_3_7 + c_4_14·a_1_22
  32. b_2_5·b_1_1·a_3_9 + b_2_5·a_4_10 + b_2_4·b_1_1·a_3_9 + a_3_7·a_3_9 + a_3_4·a_3_9
  33. b_2_5·b_1_1·a_3_7 + b_2_4·b_1_1·a_3_9 + b_2_4·a_4_10
  34. a_2_3·a_4_10
  35. a_2_3·b_4_13 + a_3_7·a_3_9 + a_3_4·a_3_9
  36. a_3_7·a_3_9 + a_3_4·a_3_9 + a_1_0·a_5_18
  37. a_2_3·b_1_14 + a_3_7·a_3_9 + a_3_72 + a_3_4·a_3_9 + a_3_4·a_3_7 + a_1_2·a_5_18
  38. a_4_10·a_3_9
  39. a_4_10·a_3_7 + a_2_3·c_4_14·a_1_2
  40. a_4_10·a_3_4
  41. b_4_13·a_3_4 + b_2_52·a_3_4 + b_2_53·a_1_0 + b_2_5·c_4_14·a_1_0
  42. b_1_12·a_5_18 + b_1_14·a_3_7 + b_4_13·a_3_7 + b_2_52·a_3_4 + b_2_53·a_1_0
       + b_2_4·b_1_12·a_3_7 + a_2_3·b_1_12·a_3_7 + c_4_14·a_1_2·b_1_12
  43. b_4_13·a_3_9 + b_2_5·a_5_18 + b_2_4·b_2_5·a_3_7 + b_2_42·a_3_7 + b_2_5·c_4_14·a_1_0
  44. a_2_3·a_5_18 + a_2_3·b_1_12·a_3_7 + a_2_3·c_4_14·a_1_2
  45. a_4_102
  46. b_4_132 + b_2_4·b_2_53 + b_2_44 + b_2_52·c_4_14
  47. a_3_9·a_5_18 + c_4_14·a_1_0·a_3_9 + a_2_3·c_4_15·a_1_2·b_1_1
       + a_2_3·c_4_14·a_1_2·b_1_1
  48. a_3_7·a_5_18 + b_1_12·a_3_72 + b_2_5·a_1_0·a_5_18 + a_2_3·b_1_13·a_3_7
       + c_4_14·a_1_2·a_3_7 + a_2_3·c_4_14·a_1_2·b_1_1
  49. a_4_10·b_4_13 + b_2_5·b_1_1·a_5_18 + b_2_4·b_1_1·a_5_18 + b_2_4·b_1_13·a_3_7
       + b_2_42·b_1_1·a_3_9 + b_2_42·a_4_10 + a_2_3·c_4_14·b_1_12 + c_4_14·a_1_0·a_3_9
  50. a_3_7·a_5_18 + a_3_4·a_5_18 + b_1_12·a_3_72 + c_4_14·a_1_2·a_3_7 + c_4_14·a_1_0·a_3_9
       + a_2_3·c_4_14·a_1_2·b_1_1
  51. b_4_13·a_5_18 + b_4_13·b_1_12·a_3_7 + b_2_4·b_4_13·a_3_7 + b_2_4·b_2_52·a_3_9
       + b_2_42·b_2_5·a_3_7 + b_2_5·c_4_14·a_3_9 + b_2_5·c_4_14·a_3_4 + b_2_52·c_4_14·a_1_0
  52. a_4_10·a_5_18 + a_2_3·c_4_14·a_3_7
  53. a_5_182 + b_1_14·a_3_72 + c_4_142·a_1_22

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 10.
  • 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_4_14, a Duflot regular element of degree 4
    2. c_4_15, a Duflot regular element of degree 4
    3. b_1_12 + b_2_5 + b_2_4, an element of degree 2
    4. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 6, 8].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → 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. a_3_4 → 0, an element of degree 3
  8. a_3_7 → 0, an element of degree 3
  9. a_3_9 → 0, an element of degree 3
  10. a_4_10 → 0, an element of degree 4
  11. b_4_13 → 0, an element of degree 4
  12. c_4_14 → c_1_04, an element of degree 4
  13. c_4_15 → c_1_14 + c_1_04, an element of degree 4
  14. a_5_18 → 0, an element of degree 5

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → 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. a_3_4 → 0, an element of degree 3
  8. a_3_7 → 0, an element of degree 3
  9. a_3_9 → 0, an element of degree 3
  10. a_4_10 → 0, an element of degree 4
  11. b_4_13 → c_1_22·c_1_32 + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32, an element of degree 4
  12. c_4_14 → c_1_2·c_1_33 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. c_4_15 → c_1_34 + c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_12·c_1_32 + c_1_14
       + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  14. a_5_18 → 0, an element of degree 5

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