Cohomology of group number 329 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 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 4, respectively.

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) · (t2  −  t  +  1) · (t3  −  t2  −  t  −  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-5,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 5:

  1. a_1_2, a nilpotent element of degree 1
  2. b_1_0, an element of degree 1
  3. b_1_1, an element of degree 1
  4. b_2_4, an element of degree 2
  5. b_2_5, an element of degree 2
  6. b_3_7, an element of degree 3
  7. b_3_8, an element of degree 3
  8. b_4_9, an element of degree 4
  9. b_4_11, an element of degree 4
  10. b_4_12, an element of degree 4
  11. c_4_13, a Duflot regular element of degree 4
  12. c_4_14, a Duflot regular element of degree 4
  13. b_5_23, an element of degree 5

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Ring relations

There are 44 minimal relations of maximal degree 10:

  1. a_1_2·b_1_0
  2. b_1_0·b_1_1
  3. a_1_22·b_1_1
  4. b_2_4·b_1_1 + a_1_23
  5. b_2_4·a_1_2 + a_1_23
  6. b_2_5·b_1_1 + a_1_23
  7. b_1_1·b_3_7
  8. a_1_2·b_3_7
  9. b_1_1·b_3_8
  10. b_1_0·b_3_8 + b_1_0·b_3_7 + b_2_42 + b_2_5·a_1_22
  11. a_1_2·b_3_8
  12. b_1_02·b_3_7 + b_4_9·b_1_0 + b_2_4·b_2_5·b_1_0 + b_2_42·b_1_0
  13. b_4_11·b_1_1 + b_4_9·b_1_1 + b_4_9·a_1_2
  14. b_4_11·b_1_0 + b_2_4·b_3_7 + b_2_4·b_2_5·b_1_0
  15. b_4_11·a_1_2 + b_4_9·a_1_2
  16. b_4_12·b_1_1 + b_4_9·a_1_2
  17. b_1_02·b_3_7 + b_4_12·b_1_0 + b_2_5·b_3_7 + b_2_42·b_1_0
  18. b_3_72 + b_2_4·b_2_5·b_1_02 + b_2_42·b_1_02 + c_4_14·b_1_02 + c_4_13·b_1_02
  19. b_2_4·b_1_0·b_3_7 + b_2_4·b_4_9 + b_2_42·b_2_5 + b_2_43
  20. b_3_82 + b_3_72 + b_4_9·b_1_02 + b_2_5·b_1_0·b_3_7 + b_2_4·b_2_5·b_1_02
       + b_2_42·b_1_02 + b_2_42·b_2_5 + b_2_43 + c_4_13·b_1_02
  21. b_3_7·b_3_8 + b_3_72 + b_2_5·b_1_0·b_3_7 + b_2_5·b_4_9 + b_2_4·b_4_11 + b_2_4·b_2_52
  22. b_2_5·b_4_11 + b_2_4·b_1_0·b_3_7 + b_2_4·b_4_12 + b_2_4·b_2_52 + b_2_43
       + b_2_52·a_1_22
  23. b_2_5·b_1_0·b_3_7 + b_2_5·b_4_9 + b_2_4·b_2_52 + b_2_42·b_2_5 + b_4_12·a_1_22
  24. b_1_1·b_5_23 + b_4_9·b_1_12 + c_4_13·a_1_2·b_1_1
  25. b_3_7·b_3_8 + b_1_0·b_5_23 + b_2_5·b_1_0·b_3_7 + b_2_4·b_1_0·b_3_7
       + b_2_4·b_2_5·b_1_02 + b_2_42·b_1_02 + b_2_42·b_2_5 + b_2_43 + b_2_52·a_1_22
  26. b_2_5·b_1_0·b_3_7 + b_2_5·b_4_9 + b_2_4·b_2_52 + b_2_42·b_2_5 + a_1_2·b_5_23
       + b_4_9·a_1_2·b_1_1 + c_4_13·a_1_22
  27. b_4_9·b_3_7 + b_2_4·b_2_5·b_3_7 + b_2_4·b_2_5·b_1_03 + b_2_42·b_3_7
       + b_2_42·b_1_03 + c_4_14·b_1_03 + c_4_13·b_1_03
  28. b_4_9·b_3_8 + b_2_4·b_2_5·b_3_8 + b_2_4·b_2_5·b_1_03 + b_2_42·b_3_8 + b_2_42·b_3_7
       + b_2_42·b_1_03 + c_4_14·b_1_03 + c_4_13·b_1_03
  29. b_4_11·b_3_7 + b_2_4·b_2_5·b_3_7 + b_2_42·b_2_5·b_1_0 + b_2_43·b_1_0
       + b_2_4·c_4_14·b_1_0 + b_2_4·c_4_13·b_1_0
  30. b_4_12·b_3_7 + b_2_4·b_2_5·b_1_03 + b_2_4·b_2_52·b_1_0 + b_2_42·b_3_7
       + b_2_42·b_1_03 + b_2_42·b_2_5·b_1_0 + c_4_14·b_1_03 + c_4_13·b_1_03
       + b_2_5·c_4_14·b_1_0 + b_2_5·c_4_13·b_1_0
  31. b_4_12·b_3_8 + b_2_5·b_5_23 + b_2_52·b_3_8 + b_2_4·b_2_5·b_3_8 + b_2_4·b_2_5·b_1_03
       + b_2_4·b_2_52·b_1_0 + b_2_42·b_3_8 + b_2_42·b_3_7 + b_2_42·b_1_03
       + b_2_42·b_2_5·b_1_0 + b_2_5·b_4_12·a_1_2 + c_4_14·b_1_03 + c_4_13·b_1_03
       + b_2_5·c_4_13·a_1_2
  32. b_4_11·b_3_8 + b_2_4·b_5_23 + b_2_42·b_3_8 + b_2_42·b_2_5·b_1_0 + b_2_43·b_1_0
       + c_4_13·a_1_23
  33. b_4_92 + b_2_4·b_2_5·b_1_04 + b_2_42·b_1_04 + b_2_42·b_2_52 + b_2_44
       + c_4_14·b_1_04 + c_4_13·b_1_14 + c_4_13·b_1_04
  34. b_4_112 + b_2_42·b_2_52 + b_2_43·b_2_5 + b_2_44 + c_4_13·b_1_14 + b_2_42·c_4_14
       + b_2_42·c_4_13
  35. b_4_9·b_4_11 + b_2_4·b_2_5·b_4_9 + b_2_42·b_4_12 + b_2_42·b_4_11 + b_2_42·b_4_9
       + b_2_42·b_2_5·b_1_02 + b_2_43·b_1_02 + c_4_13·b_1_14 + b_2_4·c_4_14·b_1_02
       + b_2_4·c_4_13·b_1_02 + c_4_13·a_1_2·b_1_13
  36. b_4_122 + b_2_4·b_2_5·b_1_04 + b_2_4·b_2_53 + b_2_42·b_1_04 + b_2_42·b_2_52
       + b_2_44 + c_4_14·b_1_04 + c_4_13·b_1_04 + b_2_52·c_4_14 + b_2_52·c_4_13
  37. b_4_11·b_4_12 + b_2_4·b_2_5·b_4_12 + b_2_42·b_4_11 + b_2_42·b_2_5·b_1_02
       + b_2_42·b_2_52 + b_2_43·b_1_02 + b_2_5·b_4_12·a_1_22 + b_2_4·c_4_14·b_1_02
       + b_2_4·c_4_13·b_1_02 + b_2_4·b_2_5·c_4_14 + b_2_4·b_2_5·c_4_13
       + c_4_13·a_1_2·b_1_13
  38. b_4_9·b_4_12 + b_4_9·b_4_11 + b_2_4·b_2_5·b_1_04 + b_2_4·b_2_5·b_4_12
       + b_2_4·b_2_5·b_4_9 + b_2_4·b_2_52·b_1_02 + b_2_42·b_1_04 + b_2_42·b_4_11
       + b_2_43·b_1_02 + b_2_43·b_2_5 + b_2_44 + c_4_14·b_1_04 + c_4_13·b_1_14
       + c_4_13·b_1_04 + b_2_5·c_4_14·b_1_02 + b_2_5·c_4_13·b_1_02
       + b_2_4·c_4_14·b_1_02 + b_2_4·c_4_13·b_1_02 + b_2_5·c_4_14·a_1_22
       + b_2_5·c_4_13·a_1_22
  39. b_3_7·b_5_23 + b_4_9·b_4_11 + b_2_4·b_2_5·b_4_9 + b_2_4·b_2_52·b_1_02
       + b_2_42·b_2_5·b_1_02 + b_2_43·b_2_5 + b_2_44 + b_2_5·b_4_12·a_1_22
       + c_4_14·b_1_0·b_3_7 + c_4_13·b_1_14 + c_4_13·b_1_0·b_3_7 + b_2_5·c_4_14·b_1_02
       + b_2_5·c_4_13·b_1_02 + b_2_42·c_4_14 + b_2_42·c_4_13 + c_4_13·a_1_2·b_1_13
       + b_2_5·c_4_14·a_1_22 + b_2_5·c_4_13·a_1_22
  40. b_3_8·b_5_23 + b_4_9·b_4_11 + b_2_5·b_4_9·b_1_02 + b_2_52·b_4_9
       + b_2_4·b_4_9·b_1_02 + b_2_4·b_2_5·b_1_04 + b_2_4·b_2_52·b_1_02 + b_2_4·b_2_53
       + b_2_42·b_1_04 + b_2_42·b_2_52 + b_2_43·b_1_02 + b_2_5·b_4_12·a_1_22
       + c_4_14·b_1_0·b_3_7 + c_4_14·b_1_04 + c_4_13·b_1_14 + c_4_13·b_1_04
       + b_2_5·c_4_13·b_1_02 + b_2_4·c_4_13·b_1_02 + c_4_13·a_1_2·b_1_13
  41. b_4_12·b_5_23 + b_2_52·b_5_23 + b_2_53·b_3_8 + b_2_4·b_2_5·b_5_23
       + b_2_4·b_2_52·b_3_8 + b_2_4·b_2_52·b_3_7 + b_2_4·b_2_52·b_1_03
       + b_2_4·b_2_53·b_1_0 + b_2_42·b_5_23 + b_2_43·b_3_7 + b_2_43·b_1_03 + b_2_44·b_1_0
       + b_2_52·b_4_12·a_1_2 + b_4_9·c_4_14·b_1_0 + b_4_9·c_4_13·b_1_0 + b_2_5·c_4_14·b_3_8
       + b_2_5·c_4_14·b_1_03 + b_2_5·c_4_13·b_3_8 + b_2_5·c_4_13·b_1_03
       + b_2_4·c_4_14·b_1_03 + b_2_4·c_4_13·b_1_03 + b_2_4·b_2_5·c_4_14·b_1_0
       + b_2_4·b_2_5·c_4_13·b_1_0 + c_4_13·a_1_2·b_1_14 + b_4_12·c_4_13·a_1_2
       + b_2_52·c_4_14·a_1_2
  42. b_4_11·b_5_23 + b_2_4·b_2_52·b_3_8 + b_2_42·b_5_23 + b_2_42·b_2_5·b_3_8
       + b_2_42·b_2_5·b_3_7 + b_2_42·b_2_52·b_1_0 + b_2_43·b_3_7 + b_2_44·b_1_0
       + c_4_13·b_1_15 + b_2_4·c_4_14·b_3_8 + b_2_4·c_4_13·b_3_8 + c_4_13·a_1_2·b_1_14
       + b_4_9·c_4_13·a_1_2
  43. b_4_9·b_5_23 + b_2_4·b_2_5·b_5_23 + b_2_4·b_2_52·b_1_03 + b_2_42·b_5_23
       + b_2_42·b_2_5·b_3_7 + b_2_43·b_3_7 + b_2_43·b_1_03 + b_2_43·b_2_5·b_1_0
       + b_2_44·b_1_0 + c_4_13·b_1_15 + b_4_9·c_4_14·b_1_0 + b_4_9·c_4_13·b_1_0
       + b_2_5·c_4_14·b_1_03 + b_2_5·c_4_13·b_1_03 + b_2_4·c_4_14·b_1_03
       + b_2_4·c_4_13·b_1_03 + b_2_4·b_2_5·c_4_14·b_1_0 + b_2_4·b_2_5·c_4_13·b_1_0
       + b_4_9·c_4_13·a_1_2
  44. b_5_232 + b_2_52·b_4_9·b_1_02 + b_2_53·b_4_9 + b_2_4·b_2_5·b_4_9·b_1_02
       + b_2_4·b_2_52·b_4_9 + b_2_4·b_2_54 + b_2_42·b_2_52·b_1_02 + b_2_42·b_2_53
       + b_2_43·b_2_52 + b_2_44·b_1_02 + b_2_44·b_2_5 + b_2_52·b_4_12·a_1_22
       + c_4_13·b_1_16 + b_4_9·c_4_14·b_1_02 + b_4_9·c_4_13·b_1_02 + b_2_5·b_4_9·c_4_14
       + b_2_5·b_4_9·c_4_13 + b_2_52·c_4_14·b_1_02 + b_2_4·b_2_5·c_4_14·b_1_02
       + b_2_4·b_2_52·c_4_14 + b_2_4·b_2_52·c_4_13 + b_2_43·c_4_14 + b_2_43·c_4_13
       + b_4_12·c_4_14·a_1_22 + b_4_12·c_4_13·a_1_22 + b_2_52·c_4_14·a_1_22
       + b_2_52·c_4_13·a_1_22 + c_4_142·b_1_02 + c_4_13·c_4_14·b_1_02
       + c_4_132·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_13, a Duflot regular element of degree 4
    2. c_4_14, a Duflot regular element of degree 4
    3. b_1_12 + b_1_02 + b_2_5, an element of degree 2
    4. b_1_02, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 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_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_7 → 0, an element of degree 3
  7. b_3_8 → 0, an element of degree 3
  8. b_4_9 → 0, an element of degree 4
  9. b_4_11 → 0, an element of degree 4
  10. b_4_12 → 0, an element of degree 4
  11. c_4_13 → c_1_14, an element of degree 4
  12. c_4_14 → c_1_14 + c_1_04, an element of degree 4
  13. b_5_23 → 0, an element of degree 5

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → c_1_2, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_7 → 0, an element of degree 3
  7. b_3_8 → 0, an element of degree 3
  8. b_4_9 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
  9. b_4_11 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
  10. b_4_12 → 0, an element of degree 4
  11. c_4_13 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  12. c_4_14 → c_1_1·c_1_23 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. b_5_23 → c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → c_1_2, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_2_4 → c_1_1·c_1_2, an element of degree 2
  5. b_2_5 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. b_3_7 → c_1_1·c_1_22 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  7. b_3_8 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  8. b_4_9 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_12·c_1_22
       + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  9. b_4_11 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_22 + c_1_0·c_1_1·c_1_22
       + c_1_02·c_1_1·c_1_2, an element of degree 4
  10. b_4_12 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_12·c_1_22
       + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32
       + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  11. c_4_13 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_13·c_1_2 + c_1_14 + c_1_0·c_1_2·c_1_32
       + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
       + c_1_02·c_1_22, an element of degree 4
  12. c_4_14 → c_1_1·c_1_23 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_13·c_1_2 + c_1_14
       + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32
       + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
  13. b_5_23 → c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_12·c_1_23
       + c_1_0·c_1_22·c_1_32 + c_1_0·c_1_23·c_1_3 + c_1_0·c_1_1·c_1_23
       + c_1_0·c_1_12·c_1_22 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3
       + c_1_02·c_1_23 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_2, 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