Cohomology of group number 259 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
    t2  −  t  +  1
    (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-6,-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 6:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. a_2_4, a nilpotent element of degree 2
  5. a_3_5, a nilpotent element of degree 3
  6. a_3_7, a nilpotent element of degree 3
  7. b_3_6, an element of degree 3
  8. a_4_10, a nilpotent element of degree 4
  9. b_4_11, an element of degree 4
  10. c_4_12, a Duflot regular element of degree 4
  11. c_4_13, a Duflot regular element of degree 4
  12. b_5_21, an element of degree 5
  13. b_6_29, an element of degree 6

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

Ring relations

There are 52 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_1_0·b_1_22
  4. a_2_4·a_1_0
  5. a_2_4·b_1_1
  6. a_2_42
  7. a_1_0·a_3_5
  8. b_1_1·a_3_5 + a_2_4·b_1_22
  9. a_1_0·a_3_7
  10. b_1_1·a_3_7
  11. a_1_0·b_3_6
  12. a_2_4·a_3_5
  13. a_2_4·a_3_7
  14. b_1_22·a_3_7 + a_2_4·b_3_6 + a_2_4·b_1_23
  15. a_4_10·a_1_0
  16. b_1_22·a_3_7 + a_4_10·b_1_1 + a_2_4·b_1_23
  17. b_4_11·a_1_0
  18. a_2_4·b_1_24 + a_3_52
  19. a_3_72
  20. a_3_5·a_3_7
  21. a_3_7·b_3_6 + a_2_4·b_1_2·b_3_6
  22. b_3_62 + b_1_1·b_1_22·b_3_6 + b_1_12·b_1_2·b_3_6 + b_1_13·b_3_6 + a_2_4·b_1_24
       + c_4_12·b_1_12
  23. a_3_5·b_3_6 + a_4_10·b_1_22
  24. a_2_4·a_4_10
  25. a_2_4·b_4_11
  26. a_1_0·b_5_21
  27. b_1_1·b_5_21 + b_1_12·b_1_2·b_3_6 + b_4_11·b_1_22 + b_4_11·b_1_1·b_1_2
       + a_2_4·b_1_2·b_3_6
  28. a_4_10·a_3_7
  29. a_2_4·b_1_22·b_3_6 + a_4_10·a_3_5
  30. a_4_10·b_3_6 + a_2_4·b_1_22·b_3_6
  31. b_4_11·a_3_7
  32. b_4_11·a_3_5
  33. a_2_4·b_5_21
  34. b_6_29·a_1_0
  35. b_1_12·b_1_22·b_3_6 + b_1_14·b_3_6 + b_6_29·b_1_1 + b_4_11·b_3_6
       + b_4_11·b_1_12·b_1_2 + a_2_4·b_1_22·b_3_6 + c_4_13·b_1_1·b_1_22 + c_4_13·b_1_13
  36. a_4_102
  37. b_1_14·b_1_2·b_3_6 + b_4_112 + c_4_13·b_1_14
  38. a_4_10·b_4_11
  39. a_3_7·b_5_21
  40. a_3_5·b_5_21 + a_4_10·b_1_2·a_3_5
  41. b_3_6·b_5_21 + b_1_1·b_1_24·b_3_6 + b_6_29·b_1_22 + b_6_29·b_1_1·b_1_2
       + b_4_11·b_1_1·b_1_23 + b_4_11·b_1_12·b_1_22 + a_4_10·b_1_24 + c_4_13·b_1_24
       + c_4_13·b_1_1·b_1_23 + c_4_13·b_1_12·b_1_22 + c_4_13·b_1_13·b_1_2
       + c_4_12·b_1_13·b_1_2
  42. a_2_4·b_6_29 + a_2_4·c_4_13·b_1_22
  43. a_4_10·b_5_21
  44. b_6_29·b_1_12·b_1_2 + b_4_11·b_5_21 + b_4_11·b_1_13·b_1_22 + b_4_112·b_1_2
       + b_4_112·b_1_1 + c_4_13·b_1_12·b_1_23 + c_4_13·b_1_13·b_1_22
       + c_4_13·b_1_14·b_1_2 + c_4_13·b_1_15
  45. b_6_29·a_3_7 + a_2_4·c_4_13·b_3_6 + a_2_4·c_4_13·b_1_23
  46. b_6_29·a_3_5 + c_4_13·b_1_22·a_3_5
  47. b_6_29·b_3_6 + b_6_29·b_1_1·b_1_22 + b_6_29·b_1_13 + b_4_11·b_5_21
       + b_4_11·b_1_1·b_1_2·b_3_6 + b_4_11·b_1_12·b_1_23 + b_4_11·b_1_14·b_1_2
       + b_4_112·b_1_2 + b_4_112·b_1_1 + c_4_13·b_1_22·b_3_6 + c_4_13·b_1_1·b_1_24
       + c_4_13·b_1_12·b_3_6 + c_4_13·b_1_13·b_1_22 + c_4_12·b_1_13·b_1_22
       + c_4_12·b_1_15 + b_4_11·c_4_12·b_1_1
  48. b_5_212 + b_6_29·b_1_1·b_1_23 + b_4_11·b_1_2·b_5_21 + b_4_11·b_1_23·b_3_6
       + b_4_11·b_1_1·b_1_22·b_3_6 + b_4_11·b_1_12·b_1_24 + b_4_112·b_1_1·b_1_2
       + c_4_13·b_1_1·b_1_25 + c_4_13·b_1_12·b_1_24 + c_4_13·b_1_15·b_1_2
       + c_4_12·b_1_14·b_1_22 + c_4_13·a_3_52
  49. b_4_11·b_1_1·b_1_22·b_3_6 + b_4_11·b_1_13·b_3_6 + b_4_11·b_6_29 + b_4_112·b_1_22
       + b_4_112·b_1_12 + c_4_13·b_1_13·b_3_6 + c_4_13·b_1_14·b_1_22
       + c_4_13·b_1_15·b_1_2 + c_4_13·b_1_16 + c_4_12·b_1_15·b_1_2
       + b_4_11·c_4_13·b_1_22 + b_4_11·c_4_13·b_1_12
  50. a_4_10·b_6_29 + a_4_10·c_4_13·b_1_22
  51. b_6_29·b_5_21 + b_4_11·b_1_24·b_3_6 + b_4_11·b_1_14·b_3_6 + b_4_11·b_6_29·b_1_1
       + c_4_13·b_1_22·b_5_21 + c_4_13·b_1_14·b_1_23 + c_4_13·b_1_15·b_1_22
       + c_4_13·b_1_16·b_1_2 + c_4_13·b_6_29·b_1_1 + c_4_12·b_1_15·b_1_22
       + b_4_11·c_4_13·b_3_6 + b_4_11·c_4_13·b_1_13 + b_4_11·c_4_12·b_1_12·b_1_2
       + c_4_132·b_1_1·b_1_22 + c_4_132·b_1_13
  52. b_6_29·b_1_16 + b_6_292 + b_4_11·b_1_23·b_5_21 + b_4_11·b_1_17·b_1_2
       + b_4_11·b_6_29·b_1_22 + b_4_11·b_6_29·b_1_12 + b_4_112·b_1_1·b_1_23
       + b_4_112·b_1_14 + c_4_13·b_1_13·b_1_25 + c_4_13·b_1_14·b_1_24
       + c_4_13·b_1_17·b_1_2 + c_4_12·b_1_14·b_1_24 + c_4_12·b_1_16·b_1_22
       + c_4_12·b_1_18 + b_4_11·c_4_13·b_1_24 + b_4_11·c_4_13·b_1_14 + b_4_112·c_4_13
       + b_4_112·c_4_12 + c_4_132·b_1_24

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 12.
  • 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_12, a Duflot regular element of degree 4
    2. c_4_13, a Duflot regular element of degree 4
    3. b_1_22 + b_1_1·b_1_2 + b_1_12, an element of degree 2
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 2, 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. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. a_3_7 → 0, an element of degree 3
  7. b_3_6 → 0, an element of degree 3
  8. a_4_10 → 0, an element of degree 4
  9. b_4_11 → 0, an element of degree 4
  10. c_4_12 → c_1_04, an element of degree 4
  11. c_4_13 → c_1_14, an element of degree 4
  12. b_5_21 → 0, an element of degree 5
  13. b_6_29 → 0, an element of degree 6

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. a_3_7 → 0, an element of degree 3
  7. b_3_6 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  8. a_4_10 → 0, 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. c_4_12 → 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
  11. c_4_13 → c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_2·c_1_3, an element of degree 4
  12. b_5_21 → c_1_1·c_1_22·c_1_32 + c_1_1·c_1_23·c_1_3 + c_1_12·c_1_2·c_1_32
       + c_1_12·c_1_22·c_1_3 + c_1_0·c_1_23·c_1_3 + c_1_02·c_1_22·c_1_3, an element of degree 5
  13. b_6_29 → c_1_1·c_1_24·c_1_3 + c_1_12·c_1_22·c_1_32 + c_1_12·c_1_23·c_1_3
       + c_1_12·c_1_24 + c_1_14·c_1_32 + c_1_14·c_1_22 + c_1_0·c_1_22·c_1_33
       + c_1_0·c_1_23·c_1_32 + c_1_0·c_1_24·c_1_3 + c_1_0·c_1_25 + c_1_0·c_1_1·c_1_24
       + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_2·c_1_33 + c_1_02·c_1_22·c_1_32
       + c_1_02·c_1_23·c_1_3 + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_23
       + c_1_02·c_1_12·c_1_22, an element of degree 6

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