Cohomology of group number 785 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) · (t6  +  t2  +  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-4,-4,-3. 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 8:

  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. 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. b_3_9, an element of degree 3
  8. b_5_17, an element of degree 5
  9. b_6_20, an element of degree 6
  10. b_6_22, an element of degree 6
  11. a_7_12, a nilpotent element of degree 7
  12. a_7_13, a nilpotent element of degree 7
  13. c_8_35, a Duflot regular element of degree 8

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Ring relations

There are 49 minimal relations of maximal degree 14:

  1. a_1_2·b_1_0
  2. b_1_0·b_1_1 + a_1_22
  3. a_1_2·b_1_1
  4. a_2_3·a_1_2
  5. b_2_5·b_1_0 + b_2_4·b_1_1
  6. b_2_5·a_1_22 + b_2_4·a_1_22 + a_2_32
  7. b_1_1·b_3_9 + a_2_3·b_2_5
  8. b_1_0·b_3_9 + a_2_3·b_2_4
  9. b_2_4·b_2_5·b_1_1 + b_2_42·b_1_1
  10. a_2_3·b_3_9 + a_1_22·b_3_9
  11. b_1_1·b_5_17 + b_2_5·b_1_14 + a_2_3·b_2_5·b_1_12 + a_2_3·b_2_52
       + a_2_3·b_2_4·b_2_5 + b_2_42·a_1_22
  12. b_1_0·b_5_17 + b_2_4·b_1_04 + b_2_42·b_1_02 + a_2_3·b_1_04
  13. b_3_92 + b_2_4·b_2_52 + b_2_42·b_2_5 + a_1_2·b_5_17 + b_2_4·a_1_2·b_3_9
       + a_2_3·b_2_4·b_2_5
  14. a_2_3·b_5_17 + a_2_3·b_2_5·b_1_13 + a_2_3·b_2_4·b_1_03 + a_2_3·b_2_42·b_1_0
  15. b_6_20·b_1_0 + b_2_42·b_1_03 + b_2_43·b_1_0 + a_2_3·b_2_4·b_1_03
       + a_2_3·b_2_42·b_1_0
  16. b_6_20·a_1_2 + b_2_43·a_1_2 + b_2_4·a_1_22·b_3_9
  17. b_6_22·b_1_1 + b_6_20·b_1_1 + b_2_5·b_1_15 + b_2_53·b_1_1 + a_2_3·b_1_15
  18. b_6_22·a_1_2 + b_2_53·a_1_2 + b_2_4·b_2_52·a_1_2 + b_2_42·b_2_5·a_1_2 + b_2_43·a_1_2
       + b_2_4·a_1_22·b_3_9
  19. b_2_4·b_6_20 + b_2_43·b_1_02 + b_2_44 + b_2_5·a_1_2·b_5_17 + b_2_52·a_1_2·b_3_9
       + a_2_3·b_2_42·b_1_02 + a_2_3·b_2_43
  20. b_2_5·b_6_22 + b_2_5·b_6_20 + b_2_52·b_1_14 + b_2_54 + b_2_4·b_2_53
       + b_2_42·b_2_52 + b_2_4·a_1_2·b_5_17 + a_2_3·b_2_5·b_1_14 + b_2_43·a_1_22
  21. b_1_1·a_7_12 + a_2_3·b_2_5·b_1_14 + a_2_3·b_2_52·b_1_12 + b_2_43·a_1_22
  22. b_1_0·a_7_12 + a_2_3·b_1_06 + a_2_3·b_6_22 + a_2_3·b_6_20 + a_2_3·b_2_5·b_1_14
       + a_2_3·b_2_53 + a_2_3·b_2_42·b_1_02 + b_2_43·a_1_22
  23. a_1_2·a_7_12 + b_2_43·a_1_22
  24. b_1_1·a_7_13 + a_2_3·b_6_20 + a_2_3·b_2_5·b_1_14 + a_2_3·b_2_42·b_1_02
       + a_2_3·b_2_43 + b_2_43·a_1_22
  25. b_1_0·a_7_13 + a_2_3·b_2_4·b_1_04 + b_2_43·a_1_22
  26. a_1_2·a_7_13 + b_2_43·a_1_22
  27. b_2_44·b_1_1 + a_1_2·b_3_9·b_5_17 + b_2_5·a_7_12 + b_2_54·a_1_2 + b_2_4·b_2_53·a_1_2
       + b_2_43·b_2_5·a_1_2 + a_2_3·b_2_52·b_1_13 + a_2_3·b_2_53·b_1_1
       + b_2_42·a_1_22·b_3_9
  28. b_6_22·b_3_9 + b_6_20·b_3_9 + b_2_53·b_3_9 + b_2_4·b_2_52·b_3_9 + b_2_42·b_2_5·b_3_9
       + b_2_44·b_1_1 + b_2_4·a_7_12 + b_2_4·b_2_53·a_1_2 + b_2_42·b_2_52·a_1_2
       + b_2_44·a_1_2 + a_2_3·b_2_52·b_1_13 + a_2_3·b_2_4·b_1_05 + a_2_3·b_2_43·b_1_0
  29. b_2_42·a_1_22·b_3_9 + a_2_3·a_7_12
  30. b_6_20·b_3_9 + b_2_43·b_3_9 + b_2_44·b_1_1 + b_2_5·a_7_13 + b_2_42·b_2_52·a_1_2
       + a_2_3·b_2_52·b_1_13 + a_2_3·b_2_43·b_1_0 + b_2_42·a_1_22·b_3_9
  31. b_2_44·b_1_1 + a_1_2·b_3_9·b_5_17 + b_2_4·a_7_13 + b_2_4·b_2_53·a_1_2
       + b_2_42·b_2_52·a_1_2 + b_2_43·b_2_5·a_1_2 + a_2_3·b_2_42·b_1_03
       + b_2_42·a_1_22·b_3_9
  32. b_2_42·a_1_22·b_3_9 + a_2_3·a_7_13
  33. b_3_9·a_7_12 + b_2_53·a_1_2·b_3_9 + b_2_4·b_2_5·a_1_2·b_5_17
       + b_2_4·b_2_52·a_1_2·b_3_9 + b_2_42·a_1_2·b_5_17 + b_2_43·a_1_2·b_3_9
       + a_2_3·b_2_43·b_2_5
  34. b_3_9·a_7_13 + b_2_52·a_1_2·b_5_17 + b_2_53·a_1_2·b_3_9 + b_2_4·b_2_5·a_1_2·b_5_17
       + b_2_4·b_2_52·a_1_2·b_3_9 + b_2_42·b_2_5·a_1_2·b_3_9 + a_2_3·b_2_43·b_2_5
  35. b_5_172 + b_2_52·b_1_16 + b_2_4·b_2_54 + b_2_42·b_1_06 + b_2_43·b_2_52
       + b_2_44·b_1_02 + b_2_52·a_1_2·b_5_17 + b_2_4·b_2_5·a_1_2·b_5_17 + c_8_35·a_1_22
  36. b_6_22·b_5_17 + b_2_5·b_6_20·b_1_13 + b_2_52·b_1_17 + b_2_53·b_5_17
       + b_2_4·b_6_22·b_1_03 + b_2_4·b_2_52·b_5_17 + b_2_42·b_6_22·b_1_0
       + b_2_42·b_2_5·b_5_17 + b_2_43·b_5_17 + b_2_44·b_1_03 + b_2_45·b_1_0
       + b_2_52·a_7_13 + b_2_4·b_2_54·a_1_2 + b_2_42·b_2_53·a_1_2 + b_2_44·b_2_5·a_1_2
       + a_2_3·b_6_22·b_1_03 + a_2_3·b_2_5·b_1_17 + a_2_3·b_2_5·b_6_20·b_1_1
       + a_2_3·b_2_52·b_1_15 + a_2_3·b_2_43·b_1_03
  37. b_6_22·b_5_17 + b_6_20·b_5_17 + b_2_52·b_1_17 + b_2_53·b_5_17
       + b_2_4·b_6_22·b_1_03 + b_2_4·b_2_52·b_5_17 + b_2_42·b_6_22·b_1_0
       + b_2_42·b_2_5·b_5_17 + b_2_43·b_1_05 + b_2_45·b_1_0 + b_2_4·a_1_2·b_3_9·b_5_17
       + b_2_42·a_7_13 + b_2_43·b_2_52·a_1_2 + a_2_3·b_6_22·b_1_03 + a_2_3·b_2_5·b_1_17
       + a_2_3·b_2_52·b_1_15 + a_2_3·b_2_53·b_1_13 + a_2_3·b_2_44·b_1_0
       + a_2_3·b_2_4·a_7_12
  38. b_6_20·b_6_22 + b_6_202 + b_2_5·b_6_20·b_1_14 + b_2_53·b_6_20
       + b_2_42·b_6_22·b_1_02 + b_2_43·b_6_22 + b_2_43·b_2_53 + b_2_44·b_1_04
       + b_2_46 + b_2_53·a_1_2·b_5_17 + b_2_54·a_1_2·b_3_9 + b_2_4·b_2_52·a_1_2·b_5_17
       + b_2_4·b_2_53·a_1_2·b_3_9 + b_2_42·b_2_5·a_1_2·b_5_17
       + b_2_42·b_2_52·a_1_2·b_3_9 + a_2_3·b_6_20·b_1_14 + a_2_3·b_2_4·b_6_22·b_1_02
       + a_2_3·b_2_42·b_6_22 + a_2_3·b_2_44·b_2_5 + b_2_45·a_1_22
  39. b_6_20·b_6_22 + b_6_202 + b_2_5·b_6_20·b_1_14 + b_2_53·b_6_20
       + b_2_42·b_6_22·b_1_02 + b_2_43·b_6_22 + b_2_43·b_2_53 + b_2_44·b_1_04
       + b_2_46 + b_5_17·a_7_12 + b_2_54·a_1_2·b_3_9 + b_2_42·b_2_5·a_1_2·b_5_17
       + b_2_42·b_2_52·a_1_2·b_3_9 + b_2_43·a_1_2·b_5_17 + b_2_43·b_2_5·a_1_2·b_3_9
       + a_2_3·b_6_20·b_1_14 + a_2_3·b_2_52·b_1_16 + a_2_3·b_2_53·b_1_14
       + a_2_3·b_2_4·b_1_08 + a_2_3·b_2_42·b_1_06 + a_2_3·b_2_44·b_1_02
       + a_2_3·b_2_45
  40. b_5_17·a_7_13 + b_2_53·a_1_2·b_5_17 + b_2_54·a_1_2·b_3_9
       + b_2_4·b_2_52·a_1_2·b_5_17 + b_2_42·b_2_5·a_1_2·b_5_17
       + b_2_42·b_2_52·a_1_2·b_3_9 + a_2_3·b_2_5·b_6_20·b_1_12 + a_2_3·b_2_52·b_1_16
       + a_2_3·b_2_42·b_1_06 + a_2_3·b_2_43·b_1_04 + b_2_45·a_1_22
  41. b_6_222 + b_6_202 + b_2_52·b_1_18 + b_2_56 + b_2_4·b_1_010
       + b_2_4·b_6_22·b_1_04 + b_2_42·b_1_08 + b_2_42·b_6_22·b_1_02 + b_2_42·b_2_54
       + b_2_44·b_1_04 + b_2_44·b_2_52 + a_2_3·b_1_010 + a_2_3·b_6_22·b_1_04
       + a_2_3·b_2_4·b_1_08 + a_2_3·b_2_42·b_1_06 + a_2_3·b_2_43·b_1_04
       + c_8_35·b_1_04
  42. b_6_202 + b_2_5·b_1_110 + b_2_5·b_6_20·b_1_14 + b_2_52·b_6_20·b_1_12
       + b_2_54·b_1_14 + b_2_55·b_1_12 + b_2_44·b_1_04 + b_2_46 + a_2_3·b_1_110
       + a_2_3·b_6_20·b_1_14 + a_2_3·b_2_53·b_1_14 + a_2_3·b_2_54·b_1_12
       + c_8_35·b_1_14
  43. b_6_20·a_7_12 + b_2_43·a_7_12 + a_2_3·b_2_5·b_6_20·b_1_13
       + a_2_3·b_2_52·b_6_20·b_1_1 + a_2_3·b_2_42·b_1_07 + a_2_3·b_2_42·b_6_22·b_1_0
       + a_2_3·b_2_45·b_1_0
  44. b_6_22·a_7_13 + b_6_20·a_7_13 + b_2_53·a_7_13 + b_2_4·b_2_52·a_7_13
       + b_2_42·b_2_5·a_7_13 + a_2_3·b_2_5·b_6_20·b_1_13 + a_2_3·b_2_52·b_1_17
       + a_2_3·b_2_4·b_6_22·b_1_03 + a_2_3·b_2_43·b_1_05 + a_2_3·b_2_44·b_1_03
  45. b_6_22·a_7_13 + b_6_22·a_7_12 + b_6_20·a_7_13 + b_6_20·a_7_12 + b_2_53·a_7_13
       + b_2_56·a_1_2 + b_2_4·b_2_55·a_1_2 + b_2_43·a_7_13 + b_2_43·b_2_53·a_1_2
       + b_2_44·b_2_52·a_1_2 + a_2_3·b_6_22·b_1_05 + a_2_3·b_2_5·b_6_20·b_1_13
       + a_2_3·b_2_53·b_1_15 + a_2_3·b_2_54·b_1_13 + a_2_3·b_2_55·b_1_1
       + a_2_3·b_2_4·b_1_09 + a_2_3·b_2_45·b_1_0 + a_2_3·b_2_42·a_7_12
       + a_2_3·c_8_35·b_1_03
  46. b_6_20·a_7_13 + b_2_43·a_7_13 + a_2_3·b_2_5·b_1_19 + a_2_3·b_2_52·b_6_20·b_1_1
       + a_2_3·b_2_54·b_1_13 + a_2_3·b_2_55·b_1_1 + a_2_3·b_2_43·b_1_05
       + a_2_3·c_8_35·b_1_13
  47. a_7_122
  48. a_7_132
  49. a_7_12·a_7_13

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 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_35, a Duflot regular element of degree 8
    2. b_1_14 + b_1_04 + b_2_52 + b_2_4·b_2_5 + b_2_42, an element of degree 4
    3. b_3_9 + b_2_5·b_1_1 + b_2_4·b_1_1 + b_2_4·b_1_0, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 8, 12].
  • 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_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. 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. b_3_9 → 0, an element of degree 3
  8. b_5_17 → 0, an element of degree 5
  9. b_6_20 → 0, an element of degree 6
  10. b_6_22 → 0, an element of degree 6
  11. a_7_12 → 0, an element of degree 7
  12. a_7_13 → 0, an element of degree 7
  13. c_8_35 → c_1_08, an element of degree 8

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 → c_1_1, 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 → c_1_22 + c_1_1·c_1_2, 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 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  9. b_6_20 → c_1_26 + c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_14·c_1_22, an element of degree 6
  10. b_6_22 → c_1_26 + c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_15·c_1_2 + c_1_02·c_1_14
       + c_1_04·c_1_12, an element of degree 6
  11. a_7_12 → 0, an element of degree 7
  12. a_7_13 → 0, an element of degree 7
  13. c_8_35 → c_1_28 + c_1_1·c_1_27 + c_1_15·c_1_23 + c_1_17·c_1_2 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + 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_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → c_1_1, 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 → 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 → c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  9. b_6_20 → c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_14·c_1_22
       + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  10. b_6_22 → c_1_26 + c_1_15·c_1_2 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  11. a_7_12 → 0, an element of degree 7
  12. a_7_13 → 0, an element of degree 7
  13. c_8_35 → c_1_28 + c_1_1·c_1_27 + c_1_14·c_1_24 + c_1_15·c_1_23 + c_1_16·c_1_22
       + c_1_17·c_1_2 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_24
       + c_1_04·c_1_13·c_1_2 + 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_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. a_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_13·c_1_22, an element of degree 5
  9. b_6_20 → c_1_16, an element of degree 6
  10. b_6_22 → c_1_26 + c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_16, an element of degree 6
  11. a_7_12 → 0, an element of degree 7
  12. a_7_13 → 0, an element of degree 7
  13. c_8_35 → c_1_28 + c_1_12·c_1_26 + c_1_16·c_1_22 + c_1_18 + 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