Cohomology of group number 812 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 2 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  +  t4  +  t  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (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 14 minimal generators of maximal degree 8:

  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. b_2_3, an element of degree 2
  5. b_2_4, an element of degree 2
  6. a_3_6, a nilpotent element of degree 3
  7. a_4_4, a nilpotent element of degree 4
  8. a_4_5, a nilpotent element of degree 4
  9. b_5_11, an element of degree 5
  10. b_5_12, an element of degree 5
  11. b_6_15, an element of degree 6
  12. a_7_9, a nilpotent element of degree 7
  13. b_8_23, an element of degree 8
  14. c_8_25, 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 61 minimal relations of maximal degree 16:

  1. a_1_0·a_1_2 + a_1_02
  2. a_1_0·b_1_1 + a_1_22
  3. a_1_2·b_1_1
  4. b_2_3·b_1_1 + b_2_4·a_1_2 + b_2_4·a_1_0
  5. a_1_0·a_3_6
  6. a_1_2·a_3_6
  7. b_2_42·a_1_2 + b_2_42·a_1_0 + b_2_3·b_2_4·a_1_2 + b_2_3·b_2_4·a_1_0
  8. a_4_4·b_1_1
  9. b_2_3·a_3_6 + a_4_4·a_1_0
  10. b_2_3·a_3_6 + a_4_4·a_1_2
  11. b_1_12·a_3_6 + a_4_5·b_1_1 + b_2_4·a_3_6
  12. b_2_3·a_3_6 + a_4_5·a_1_0
  13. a_4_5·a_1_2
  14. a_3_62
  15. b_1_1·b_5_11
  16. b_1_1·b_5_12 + b_2_42·b_1_12 + a_1_2·b_5_11 + a_4_5·b_1_12 + b_2_4·b_1_1·a_3_6
       + b_2_4·a_4_5 + b_2_3·a_4_5 + b_2_32·a_1_02
  17. a_1_0·b_5_12 + b_2_4·a_4_4 + b_2_3·a_4_4 + b_2_32·a_1_02
  18. a_1_2·b_5_12 + a_1_2·b_5_11 + a_1_0·b_5_11 + b_2_4·a_4_4 + b_2_3·a_4_4 + b_2_32·a_1_02
  19. a_4_4·a_3_6
  20. a_4_5·a_3_6
  21. b_6_15·a_1_0 + b_2_32·b_2_4·a_1_0 + b_2_33·a_1_0
  22. b_6_15·a_1_2 + b_2_32·b_2_4·a_1_2 + b_2_33·a_1_2 + b_2_3·a_4_4·a_1_0
  23. a_4_42
  24. a_4_4·a_4_5
  25. a_4_52
  26. a_3_6·b_5_11
  27. a_3_6·b_5_12 + b_2_42·b_1_1·a_3_6
  28. b_2_3·b_6_15 + b_2_33·b_2_4 + b_2_34 + b_2_4·a_1_0·b_5_11 + b_2_3·a_1_0·b_5_11
       + b_2_3·b_2_4·a_4_4 + b_2_32·a_4_5 + b_2_33·a_1_02
  29. a_1_0·a_7_9
  30. a_1_2·a_7_9
  31. a_4_5·b_5_11 + b_2_32·b_2_42·a_1_0 + b_2_33·b_2_4·a_1_0 + b_2_32·a_4_4·a_1_0
  32. a_4_4·b_5_12 + b_2_3·b_2_43·a_1_0 + b_2_33·b_2_4·a_1_0 + b_2_3·b_2_4·a_4_4·a_1_0
  33. a_4_5·b_5_12 + a_4_4·b_5_11 + b_2_42·a_4_5·b_1_1 + b_2_3·b_2_4·a_4_4·a_1_0
       + b_2_32·a_4_4·a_1_0
  34. a_4_4·b_5_11 + b_2_3·a_7_9 + b_2_3·b_2_43·a_1_0 + b_2_32·b_2_42·a_1_0
       + b_2_33·b_2_4·a_1_0 + b_2_32·a_4_4·a_1_0
  35. b_1_12·a_7_9 + b_6_15·a_3_6 + b_2_4·a_4_5·b_1_13 + b_2_3·b_2_4·a_4_4·a_1_0
       + b_2_32·a_4_4·a_1_0
  36. b_8_23·b_1_1 + b_6_15·b_1_13 + b_2_4·b_6_15·b_1_1 + b_2_42·b_1_15 + b_6_15·a_3_6
       + b_2_4·a_4_5·b_1_13 + b_2_33·b_2_4·a_1_2 + b_2_33·b_2_4·a_1_0
       + b_2_3·b_2_4·a_4_4·a_1_0 + b_2_32·a_4_4·a_1_0
  37. b_8_23·a_1_0 + a_4_4·b_5_11 + b_2_32·b_2_42·a_1_0 + b_2_32·a_4_4·a_1_0
  38. b_8_23·a_1_2 + a_4_4·b_5_11 + b_2_32·b_2_42·a_1_0 + b_2_33·b_2_4·a_1_2
       + b_2_33·b_2_4·a_1_0
  39. a_4_4·b_6_15 + b_2_32·b_2_4·a_4_4 + b_2_33·a_4_4
  40. a_3_6·a_7_9
  41. b_6_15·b_1_1·a_3_6 + a_4_5·b_6_15 + b_2_4·b_1_1·a_7_9 + b_2_42·a_4_5·b_1_12
       + b_2_32·b_2_4·a_4_5 + b_2_33·a_4_5
  42. b_5_122 + b_5_112 + b_2_44·b_1_12 + b_2_3·b_2_44 + b_2_33·b_2_42
       + b_2_42·a_1_0·b_5_11 + b_2_43·a_4_4 + b_2_3·b_2_42·a_4_5 + b_2_32·b_2_4·a_4_5
       + b_2_32·b_2_4·a_4_4 + c_8_25·a_1_02
  43. b_5_112 + b_2_32·b_2_43 + b_2_34·b_2_4 + b_2_3·b_2_42·a_4_5 + b_2_33·a_4_5
       + b_2_34·a_1_02 + c_8_25·a_1_22
  44. b_5_122 + b_5_11·b_5_12 + b_2_4·b_8_23 + b_2_4·b_6_15·b_1_12 + b_2_42·b_6_15
       + b_2_43·b_1_14 + b_2_44·b_1_12 + b_2_3·b_8_23 + b_2_3·b_2_44 + b_2_32·b_2_43
       + b_2_33·b_2_42 + b_2_34·b_2_4 + b_6_15·b_1_1·a_3_6 + a_4_5·b_6_15
       + b_2_42·a_1_0·b_5_11 + b_2_42·a_4_5·b_1_12 + b_2_3·b_2_42·a_4_5
       + b_2_32·b_2_4·a_4_4 + b_2_34·a_1_02
  45. a_4_4·a_7_9 + b_2_32·b_2_4·a_4_4·a_1_0
  46. b_6_15·b_5_11 + b_2_32·b_2_4·b_5_11 + b_2_33·b_5_11 + b_2_3·b_2_4·a_7_9
       + b_2_33·b_2_42·a_1_0 + b_2_32·b_2_4·a_4_4·a_1_0 + b_2_33·a_4_4·a_1_0
  47. b_6_15·b_5_12 + b_2_42·b_6_15·b_1_1 + b_2_32·b_2_4·b_5_12 + b_2_33·b_5_12
       + a_4_5·b_6_15·b_1_1 + b_2_42·a_7_9 + b_2_43·a_4_5·b_1_1 + b_2_45·a_1_0
       + b_2_32·b_2_43·a_1_0 + b_2_33·b_2_42·a_1_0 + b_2_33·a_4_4·a_1_0
  48. a_4_5·a_7_9 + b_2_32·b_2_4·a_4_4·a_1_0
  49. b_8_23·a_3_6 + a_4_5·b_6_15·b_1_1 + b_2_42·a_4_5·b_1_13 + b_2_43·a_4_5·b_1_1
       + b_2_44·a_3_6 + b_2_32·b_2_4·a_4_4·a_1_0
  50. b_5_11·a_7_9 + b_2_43·a_1_0·b_5_11 + b_2_3·b_2_43·a_4_5 + b_2_3·b_2_43·a_4_4
       + b_2_33·b_2_4·a_4_5 + b_2_33·b_2_4·a_4_4
  51. b_5_12·a_7_9 + b_2_42·b_1_1·a_7_9 + b_2_43·a_1_0·b_5_11 + b_2_44·a_4_4
       + b_2_32·b_2_42·a_4_5 + b_2_33·b_2_4·a_4_5 + b_2_33·b_2_4·a_4_4
  52. b_6_152 + b_2_42·b_1_18 + b_2_42·b_6_15·b_1_12 + b_2_44·b_1_14
       + b_2_45·b_1_12 + b_2_34·b_2_42 + b_2_36 + b_2_4·a_4_5·b_1_16
       + b_2_42·a_4_5·b_1_14 + b_2_43·a_4_5·b_1_12 + c_8_25·b_1_14
  53. a_4_4·b_8_23 + b_2_3·b_2_43·a_4_5 + b_2_32·b_2_42·a_4_4 + b_2_33·b_2_4·a_4_5
  54. a_4_5·b_8_23 + a_4_5·b_6_15·b_1_12 + b_2_4·a_4_5·b_6_15 + b_2_42·a_4_5·b_1_14
       + b_2_32·b_2_42·a_4_4 + b_2_33·b_2_4·a_4_5 + b_2_33·b_2_4·a_4_4
  55. b_6_15·a_7_9 + b_2_4·a_4_5·b_6_15·b_1_1 + b_2_42·b_6_15·a_3_6
       + b_2_42·a_4_5·b_1_15 + b_2_43·a_4_5·b_1_13 + b_2_45·a_3_6 + b_2_32·b_2_4·a_7_9
       + b_2_33·a_7_9 + b_2_33·b_2_4·a_4_4·a_1_0 + a_4_5·c_8_25·b_1_1 + b_2_4·c_8_25·a_3_6
  56. b_8_23·b_5_11 + b_2_32·b_2_42·b_5_12 + b_2_32·b_2_42·b_5_11 + b_2_33·b_2_4·b_5_12
       + b_2_3·b_2_45·a_1_0 + b_2_32·b_2_4·a_7_9 + b_2_32·b_2_44·a_1_0 + b_2_33·a_7_9
       + b_2_33·b_2_43·a_1_0 + b_2_35·b_2_4·a_1_0 + b_2_34·a_4_4·a_1_0
       + a_4_4·c_8_25·a_1_0
  57. b_8_23·b_5_12 + b_2_42·b_6_15·b_1_13 + b_2_43·b_6_15·b_1_1 + b_2_44·b_1_15
       + b_2_3·b_2_43·b_5_11 + b_2_32·b_2_42·b_5_12 + b_2_33·b_2_4·b_5_11
       + a_4_5·b_6_15·b_1_13 + b_2_4·a_4_5·b_6_15·b_1_1 + b_2_42·a_4_5·b_1_15
       + b_2_43·a_7_9 + b_2_43·a_4_5·b_1_13 + b_2_45·a_3_6 + b_2_46·a_1_0
       + b_2_3·b_2_45·a_1_0 + b_2_32·b_2_4·a_7_9 + b_2_32·b_2_44·a_1_0
       + b_2_35·b_2_4·a_1_0 + b_2_33·b_2_4·a_4_4·a_1_0
  58. a_7_92
  59. b_6_15·b_8_23 + b_2_42·b_1_110 + b_2_43·b_1_18 + b_2_43·b_6_15·b_1_12
       + b_2_44·b_1_16 + b_2_46·b_1_12 + b_2_32·b_2_4·b_8_23 + b_2_33·b_8_23
       + b_2_4·a_4_5·b_1_18 + b_2_4·a_4_5·b_6_15·b_1_12 + b_2_42·a_4_5·b_1_16
       + b_2_42·a_4_5·b_6_15 + b_2_43·b_1_1·a_7_9 + b_2_43·a_4_5·b_1_14
       + b_2_45·b_1_1·a_3_6 + b_2_3·b_2_44·a_4_4 + b_2_32·b_2_43·a_4_5 + c_8_25·b_1_16
       + b_2_4·c_8_25·b_1_14 + a_4_5·c_8_25·b_1_12 + b_2_4·c_8_25·b_1_1·a_3_6
  60. b_8_23·a_7_9 + b_2_4·a_4_5·b_6_15·b_1_13 + b_2_42·a_4_5·b_1_17
       + b_2_42·a_4_5·b_6_15·b_1_1 + b_2_43·b_6_15·a_3_6 + b_2_43·a_4_5·b_1_15
       + b_2_44·a_4_5·b_1_13 + b_2_45·a_4_5·b_1_1 + b_2_3·b_2_43·a_7_9
       + b_2_3·b_2_46·a_1_0 + b_2_32·b_2_45·a_1_0 + b_2_33·b_2_4·a_7_9
       + b_2_34·b_2_43·a_1_0 + b_2_34·b_2_4·a_4_4·a_1_0 + a_4_5·c_8_25·b_1_13
  61. b_8_232 + b_2_42·b_1_112 + b_2_42·b_6_15·b_1_16 + b_2_44·b_1_18
       + b_2_44·b_6_15·b_1_12 + b_2_45·b_1_16 + b_2_46·b_1_14 + b_2_47·b_1_12
       + b_2_33·b_2_45 + b_2_35·b_2_43 + b_2_36·b_2_42 + b_2_4·a_4_5·b_1_110
       + b_2_42·a_4_5·b_1_18 + b_2_44·a_4_5·b_1_14 + b_2_45·a_4_5·b_1_12
       + b_2_32·b_2_44·a_4_4 + b_2_34·b_2_42·a_4_4 + c_8_25·b_1_18
       + b_2_42·c_8_25·b_1_14

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_25, a Duflot regular element of degree 8
    2. b_1_14 + b_2_42 + b_2_3·b_2_4 + b_2_32, an element of degree 4
    3. b_2_4, 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_2 → 0, an element of degree 1
  3. b_1_1 → 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. a_3_6 → 0, an element of degree 3
  7. a_4_4 → 0, an element of degree 4
  8. a_4_5 → 0, an element of degree 4
  9. b_5_11 → 0, an element of degree 5
  10. b_5_12 → 0, an element of degree 5
  11. b_6_15 → 0, an element of degree 6
  12. a_7_9 → 0, an element of degree 7
  13. b_8_23 → 0, an element of degree 8
  14. c_8_25 → 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_2 → 0, an element of degree 1
  3. b_1_1 → c_1_1, an element of degree 1
  4. b_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. a_3_6 → 0, an element of degree 3
  7. a_4_4 → 0, an element of degree 4
  8. a_4_5 → 0, an element of degree 4
  9. b_5_11 → 0, an element of degree 5
  10. b_5_12 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  11. b_6_15 → c_1_1·c_1_25 + c_1_13·c_1_23 + 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
  12. a_7_9 → 0, an element of degree 7
  13. b_8_23 → c_1_1·c_1_27 + c_1_12·c_1_26 + 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_14·c_1_22 + c_1_02·c_1_15·c_1_2
       + c_1_02·c_1_16 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_13·c_1_2
       + c_1_04·c_1_14, an element of degree 8
  14. c_8_25 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23
       + 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_2 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_2_3 → c_1_12, an element of degree 2
  5. b_2_4 → c_1_22, an element of degree 2
  6. a_3_6 → 0, an element of degree 3
  7. a_4_4 → 0, an element of degree 4
  8. a_4_5 → 0, an element of degree 4
  9. b_5_11 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  10. b_5_12 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  11. b_6_15 → c_1_14·c_1_22 + c_1_16, an element of degree 6
  12. a_7_9 → 0, an element of degree 7
  13. b_8_23 → c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22, an element of degree 8
  14. c_8_25 → c_1_28 + c_1_14·c_1_24 + 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