Cohomology of group number 25 of order 64

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

General information on the group

  • The group has 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is 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
    t6  −  t5  −  t4  +  2·t3  −  2·t2  +  t  −  1
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-5,-3,-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_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_0, a nilpotent element of degree 2
  4. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_3, an element of degree 2
  7. b_3_4, an element of degree 3
  8. b_3_5, an element of degree 3
  9. a_5_4, a nilpotent element of degree 5
  10. b_5_9, an element of degree 5
  11. a_6_6, a nilpotent element of degree 6
  12. b_6_11, an element of degree 6
  13. c_8_17, a Duflot regular element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

Ring relations

There are 53 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_0
  5. a_2_1·a_1_1
  6. a_2_1·a_1_0 + a_2_0·a_1_1
  7. b_2_3·a_1_0 + b_2_2·a_1_1 + a_2_0·a_1_1
  8. a_2_02
  9. a_2_0·a_2_1
  10. a_2_12
  11. a_1_1·b_3_4 + a_2_0·b_2_3
  12. a_1_0·b_3_4 + a_2_0·b_2_2
  13. a_1_1·b_3_5 + a_2_1·b_2_3
  14. a_1_0·b_3_5 + a_2_1·b_2_2
  15. b_2_2·b_2_3·a_1_1 + b_2_22·a_1_1
  16. a_2_0·b_3_4
  17. a_2_1·b_3_4 + a_2_0·b_3_5
  18. b_2_32·a_1_1 + a_2_1·b_3_5
  19. b_3_42 + b_2_2·b_2_32 + b_2_22·b_2_3 + a_2_0·b_2_2·b_2_3 + a_2_0·b_2_22
  20. b_3_52 + b_2_33 + a_2_1·b_2_32 + a_2_1·b_2_2·b_2_3 + a_2_0·b_2_22
  21. a_1_1·a_5_4
  22. a_2_1·b_2_2·b_2_3 + a_2_1·b_2_22 + a_1_0·a_5_4
  23. a_1_1·b_5_9 + a_2_1·b_2_32 + a_2_1·b_2_2·b_2_3 + a_2_0·b_2_32 + a_2_0·b_2_2·b_2_3
  24. a_1_0·b_5_9 + a_2_1·b_2_2·b_2_3 + a_2_1·b_2_22
  25. a_2_1·a_5_4
  26. b_2_3·b_5_9 + b_2_32·b_3_5 + b_2_32·b_3_4 + b_2_2·b_2_3·b_3_4 + b_2_22·b_3_5
       + b_2_2·a_5_4 + a_2_0·b_2_3·b_3_5 + a_2_0·b_2_2·b_3_5 + a_2_0·a_5_4
  27. a_2_0·b_5_9 + a_2_0·b_2_3·b_3_5 + a_2_0·b_2_2·b_3_5
  28. a_2_1·b_5_9 + a_2_1·b_2_3·b_3_5 + a_2_1·b_2_2·b_3_5 + a_2_0·b_2_3·b_3_5
       + a_2_0·b_2_2·b_3_5 + a_2_0·a_5_4
  29. a_6_6·a_1_1
  30. a_6_6·a_1_0 + a_2_0·a_5_4
  31. b_6_11·a_1_1 + a_2_1·b_2_3·b_3_5 + a_2_1·b_2_2·b_3_5 + a_2_0·b_2_3·b_3_5
       + a_2_0·b_2_2·b_3_5 + a_2_0·a_5_4
  32. b_6_11·a_1_0 + a_2_0·a_5_4
  33. b_3_5·a_5_4 + b_2_3·a_6_6 + a_2_1·b_2_33 + a_2_0·b_2_33 + a_2_0·b_2_23
  34. b_3_5·b_5_9 + b_2_3·b_3_4·b_3_5 + b_2_34 + b_2_2·b_3_4·b_3_5 + b_2_22·b_2_32
       + b_2_2·a_6_6 + a_2_1·b_2_33 + a_2_0·b_2_33 + a_2_0·b_2_22·b_2_3
  35. a_2_0·a_6_6
  36. a_2_1·a_6_6
  37. b_2_3·b_3_4·b_3_5 + b_2_3·b_6_11 + b_2_34 + b_2_2·b_3_4·b_3_5 + b_2_22·b_2_32
       + b_3_5·a_5_4 + b_3_4·a_5_4 + a_2_1·b_2_33 + a_2_0·b_2_33 + a_2_0·b_2_22·b_2_3
       + a_2_0·b_2_23
  38. b_3_5·b_5_9 + b_3_4·b_5_9 + b_2_34 + b_2_2·b_6_11 + b_2_22·b_2_32 + a_2_1·b_2_33
       + a_2_0·b_2_33 + b_2_2·a_1_0·a_5_4
  39. a_2_0·b_6_11 + a_2_0·b_2_33 + a_2_0·b_2_22·b_2_3
  40. a_2_1·b_6_11 + a_2_1·b_2_33 + a_2_1·b_2_23 + a_2_0·b_2_33 + a_2_0·b_2_22·b_2_3
       + b_2_2·a_1_0·a_5_4
  41. a_6_6·b_3_5 + b_2_32·a_5_4 + a_2_1·b_2_32·b_3_5 + a_2_0·b_2_32·b_3_5
       + a_2_0·b_2_22·b_3_5
  42. b_6_11·b_3_4 + b_2_33·b_3_4 + b_2_2·b_2_32·b_3_5 + b_2_22·b_2_3·b_3_4 + b_2_23·b_3_5
       + a_6_6·b_3_4 + b_2_2·b_2_3·a_5_4 + b_2_22·a_5_4 + a_2_0·b_2_2·a_5_4
  43. b_6_11·b_3_5 + b_2_33·b_3_5 + b_2_33·b_3_4 + b_2_2·b_2_32·b_3_4 + b_2_22·b_2_3·b_3_5
       + a_6_6·b_3_4 + b_2_32·a_5_4 + a_2_1·b_2_32·b_3_5 + a_2_0·b_2_32·b_3_5
       + a_2_0·b_2_22·b_3_5 + a_2_0·b_2_2·a_5_4
  44. a_2_0·b_2_23·b_2_3 + a_2_0·b_2_24 + a_5_42
  45. b_5_92 + b_2_35 + b_2_2·b_2_34 + b_2_22·b_2_33 + b_2_23·b_2_32
       + a_2_1·b_2_34 + a_2_1·b_2_24 + a_2_0·b_2_23·b_2_3 + a_2_0·b_2_24
       + b_2_22·a_1_0·a_5_4
  46. a_5_4·b_5_9 + b_2_3·b_3_4·a_5_4 + b_2_32·a_6_6 + b_2_2·b_3_4·a_5_4 + b_2_22·a_6_6
       + a_2_1·b_2_34 + a_2_1·b_2_24 + a_2_0·b_2_34 + a_2_0·b_2_23·b_2_3
       + b_2_22·a_1_0·a_5_4
  47. a_6_6·a_5_4 + a_2_0·b_2_22·a_5_4
  48. b_6_11·a_5_4 + a_6_6·b_5_9 + a_2_1·b_2_33·b_3_5 + a_2_1·b_2_23·b_3_5
  49. b_6_11·b_5_9 + b_2_34·b_3_5 + b_2_2·b_2_33·b_3_5 + b_2_22·b_2_32·b_3_5
       + b_2_23·b_2_3·b_3_5 + a_6_6·b_5_9 + b_2_3·a_6_6·b_3_4 + b_2_2·a_6_6·b_3_4
       + a_2_0·b_2_33·b_3_5 + a_2_0·b_2_23·b_3_5
  50. a_6_6·b_5_9 + b_2_3·a_6_6·b_3_4 + b_2_33·a_5_4 + b_2_2·a_6_6·b_3_4
       + b_2_22·b_2_3·a_5_4 + a_2_1·b_2_33·b_3_5 + a_2_1·b_2_23·b_3_5 + a_2_0·b_2_33·b_3_5
       + a_2_0·b_2_23·b_3_5 + a_2_0·b_2_22·a_5_4 + a_2_0·c_8_17·a_1_1
  51. a_6_62
  52. b_6_112 + b_2_36 + b_2_2·b_2_35 + b_2_22·b_2_34 + b_2_23·b_2_33
  53. a_6_6·b_6_11 + b_2_32·b_3_4·a_5_4 + b_2_33·a_6_6 + b_2_2·b_2_3·b_3_4·a_5_4
       + b_2_22·b_2_3·a_6_6 + a_2_0·b_2_35 + a_2_0·b_2_25 + b_2_2·a_5_42

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

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_8_17, a Duflot regular element of degree 8
    2. b_2_32 + b_2_2·b_2_3 + b_2_22, an element of degree 4
    3. b_3_4, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 9, 12].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 2.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

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_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. b_3_4 → 0, an element of degree 3
  8. b_3_5 → 0, an element of degree 3
  9. a_5_4 → 0, an element of degree 5
  10. b_5_9 → 0, an element of degree 5
  11. a_6_6 → 0, an element of degree 6
  12. b_6_11 → 0, an element of degree 6
  13. c_8_17 → 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_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. b_2_2 → c_1_22, an element of degree 2
  6. b_2_3 → c_1_12, an element of degree 2
  7. b_3_4 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_3_5 → c_1_13, an element of degree 3
  9. a_5_4 → 0, an element of degree 5
  10. b_5_9 → c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_15, an element of degree 5
  11. a_6_6 → 0, an element of degree 6
  12. b_6_11 → c_1_13·c_1_23 + c_1_14·c_1_22 + c_1_15·c_1_2 + c_1_16, an element of degree 6
  13. c_8_17 → c_1_12·c_1_26 + c_1_15·c_1_23 + c_1_16·c_1_22 + 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 64


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