Cohomology of group number 252 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 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    ( − 1) · (t5  −  t4  −  t2  −  1)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-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_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_1_2, an element of degree 1
  4. b_2_4, an element of degree 2
  5. b_2_5, an element of degree 2
  6. a_3_7, a nilpotent element of degree 3
  7. a_3_3, a nilpotent element of degree 3
  8. a_3_5, a nilpotent element of degree 3
  9. b_4_13, an element of degree 4
  10. b_4_14, an element of degree 4
  11. c_4_15, a Duflot regular element of degree 4
  12. c_4_16, a Duflot regular element of degree 4
  13. a_5_16, a nilpotent element of degree 5

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

Ring relations

There are 44 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_13
  4. a_1_1·b_1_22 + a_1_0·b_1_22 + b_2_4·a_1_1
  5. a_1_0·b_1_22 + b_2_5·a_1_1
  6. a_1_0·b_1_22 + b_2_5·a_1_0
  7. b_1_24 + b_2_52 + b_2_4·b_1_22 + b_2_4·a_1_0·b_1_2 + b_2_4·a_1_12
  8. a_1_1·a_3_7 + b_2_4·a_1_12
  9. a_1_0·a_3_7
  10. a_1_1·a_3_3 + b_2_4·a_1_12
  11. b_2_4·a_1_0·b_1_2 + a_1_0·a_3_3
  12. b_2_4·a_1_0·b_1_2 + a_1_0·a_3_5
  13. b_1_22·a_3_3 + b_1_22·a_3_7 + b_2_5·a_3_5 + b_2_5·a_3_3 + b_2_4·a_1_12·b_1_2
  14. b_1_22·a_3_7 + b_2_5·a_3_7 + b_2_42·a_1_1 + a_1_12·a_3_5
  15. b_1_22·a_3_5 + b_1_22·a_3_3 + b_1_22·a_3_7 + b_2_5·a_3_3 + b_2_4·a_3_5 + b_2_4·a_3_3
       + b_2_42·a_1_1
  16. b_1_22·a_3_7 + b_4_13·a_1_1 + b_2_52·a_1_0 + b_2_4·a_1_12·b_1_2
  17. b_1_22·a_3_7 + b_4_13·a_1_0 + b_2_52·a_1_0 + b_2_4·a_3_7
  18. b_4_14·a_1_1 + b_2_5·a_3_7 + b_2_52·a_1_0 + a_1_1·b_1_2·a_3_5
  19. b_4_14·a_1_0 + b_2_5·a_3_7 + b_2_52·a_1_0
  20. a_3_72 + b_2_42·a_1_12
  21. a_3_32 + b_2_4·a_1_0·a_3_3 + b_2_42·a_1_12
  22. b_2_4·b_1_2·a_3_7 + b_2_42·a_1_1·b_1_2 + a_3_7·a_3_3 + b_2_42·a_1_12
       + a_1_12·b_1_2·a_3_5
  23. b_2_4·b_1_2·a_3_7 + b_2_42·a_1_1·b_1_2 + a_3_3·a_3_5 + a_3_32 + a_3_7·a_3_5
       + b_2_42·a_1_12
  24. b_2_4·b_1_2·a_3_7 + b_2_42·a_1_1·b_1_2 + a_3_3·a_3_5 + a_3_32 + a_3_7·a_3_3
       + b_2_4·a_1_1·a_3_5
  25. b_2_4·b_1_2·a_3_7 + b_2_42·a_1_1·b_1_2 + a_3_52 + a_3_3·a_3_5 + a_3_7·a_3_3
       + b_2_42·a_1_12 + c_4_15·a_1_12
  26. b_4_13·b_1_22 + b_2_5·b_4_14 + b_2_4·b_2_5·b_1_22 + b_2_4·b_2_52
       + b_2_42·b_1_22 + b_2_5·b_1_2·a_3_3 + b_2_52·a_1_0·b_1_2 + b_2_4·b_1_2·a_3_7
       + a_3_3·a_3_5
  27. b_4_14·b_1_22 + b_2_5·b_4_13 + b_2_4·b_4_14 + b_2_4·b_2_5·b_1_22 + b_2_4·b_2_52
       + b_2_5·b_1_2·a_3_5 + b_2_5·b_1_2·a_3_3 + b_2_5·b_1_2·a_3_7 + b_2_52·a_1_0·b_1_2
       + b_2_4·b_1_2·a_3_3 + b_2_42·a_1_1·b_1_2 + a_3_32
  28. a_3_52 + a_3_3·a_3_5 + a_1_1·a_5_16 + b_2_42·a_1_12
  29. a_3_7·a_3_3 + a_1_0·a_5_16 + b_2_42·a_1_12
  30. b_4_13·a_3_7 + b_2_53·a_1_0 + b_2_42·a_3_7 + b_2_42·a_1_12·b_1_2
       + b_2_4·a_1_12·a_3_5 + b_2_5·c_4_16·a_1_0 + b_2_4·c_4_16·a_1_0
  31. b_4_14·a_3_3 + b_4_13·a_3_5 + b_4_13·a_3_3 + b_2_53·a_1_0 + b_2_4·b_2_5·a_3_5
       + b_2_42·a_3_5 + b_2_42·a_3_3 + b_2_42·a_1_12·b_1_2 + b_2_4·a_1_12·a_3_5
       + b_2_5·c_4_16·a_1_0
  32. b_4_14·a_3_7 + b_2_53·a_1_0 + b_2_4·a_1_1·b_1_2·a_3_5 + b_2_5·c_4_16·a_1_0
  33. b_4_13·a_3_5 + b_4_13·a_3_3 + b_2_5·a_5_16 + b_2_52·a_3_5 + b_2_52·a_3_7
       + b_2_4·b_2_5·a_3_5 + b_2_42·a_3_5 + b_2_42·a_3_3 + b_2_4·a_1_1·b_1_2·a_3_5
       + b_2_42·a_1_12·b_1_2 + b_2_5·c_4_16·a_1_0
  34. b_1_22·a_5_16 + b_4_14·a_3_5 + b_4_13·a_3_5 + b_4_13·a_3_3 + b_2_52·a_3_5
       + b_2_52·a_3_7 + b_2_53·a_1_0 + b_2_4·b_2_5·a_3_3 + b_2_42·a_3_5 + b_2_42·a_3_3
       + b_2_43·a_1_1 + b_2_4·a_1_1·b_1_2·a_3_5 + b_2_4·c_4_15·a_1_1 + c_4_15·a_1_12·b_1_2
  35. b_4_14·a_3_5 + b_4_13·a_3_5 + b_2_4·a_5_16 + b_2_4·b_2_5·a_3_5 + b_2_42·a_3_5
       + b_2_42·a_3_3 + b_2_4·a_1_1·b_1_2·a_3_5 + b_2_4·c_4_15·a_1_1 + b_2_4·c_4_15·a_1_0
       + c_4_15·a_1_12·b_1_2
  36. b_4_132 + b_2_52·b_4_13 + b_2_54 + b_2_4·b_2_5·b_4_14 + b_2_4·b_2_52·b_1_22
       + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22 + b_2_52·b_1_2·a_3_3
       + b_2_4·b_2_5·b_1_2·a_3_5 + b_2_42·b_1_2·a_3_3 + b_2_42·a_1_1·a_3_5
       + b_2_42·a_1_0·a_3_3 + b_2_4·a_1_12·b_1_2·a_3_5 + b_2_52·c_4_16
       + b_2_4·c_4_16·b_1_22 + b_2_42·c_4_16 + c_4_16·a_1_0·a_3_3 + b_2_4·c_4_16·a_1_12
  37. b_4_142 + b_2_52·b_4_13 + b_2_54 + b_2_52·b_1_2·a_3_3 + b_2_42·a_1_1·a_3_5
       + b_2_52·c_4_16 + b_2_4·c_4_15·a_1_12
  38. b_4_13·b_4_14 + b_4_132 + b_2_52·b_4_14 + b_2_52·b_4_13 + b_2_53·b_1_22 + b_2_54
       + b_2_42·b_4_14 + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52
       + b_4_13·b_1_2·a_3_3 + b_2_52·b_1_2·a_3_5 + b_2_52·b_1_2·a_3_3
       + b_2_4·b_2_5·b_1_2·a_3_5 + a_3_3·a_5_16 + b_2_5·c_4_16·b_1_22 + b_2_52·c_4_16
       + b_2_4·c_4_16·b_1_22 + b_2_4·b_2_5·c_4_16 + b_2_42·c_4_16 + c_4_16·a_1_0·a_3_3
       + c_4_15·a_1_0·a_3_3 + b_2_4·c_4_16·a_1_12 + b_2_4·c_4_15·a_1_12
  39. b_4_13·b_4_14 + b_4_132 + b_2_52·b_4_14 + b_2_52·b_4_13 + b_2_53·b_1_22 + b_2_54
       + b_2_42·b_4_14 + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52
       + b_4_13·b_1_2·a_3_3 + b_2_52·b_1_2·a_3_5 + b_2_52·b_1_2·a_3_3
       + b_2_4·b_2_5·b_1_2·a_3_5 + a_3_7·a_5_16 + b_2_5·c_4_16·b_1_22 + b_2_52·c_4_16
       + b_2_4·c_4_16·b_1_22 + b_2_4·b_2_5·c_4_16 + b_2_42·c_4_16 + b_2_4·c_4_16·a_1_12
       + b_2_4·c_4_15·a_1_12
  40. b_4_13·b_4_14 + b_4_132 + b_2_52·b_4_14 + b_2_52·b_4_13 + b_2_53·b_1_22 + b_2_54
       + b_2_42·b_4_14 + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52
       + b_4_13·b_1_2·a_3_3 + b_2_52·b_1_2·a_3_5 + b_2_52·b_1_2·a_3_3
       + b_2_4·b_2_5·b_1_2·a_3_5 + b_2_4·a_1_0·a_5_16 + b_2_43·a_1_12
       + b_2_4·a_1_12·b_1_2·a_3_5 + b_2_5·c_4_16·b_1_22 + b_2_52·c_4_16
       + b_2_4·c_4_16·b_1_22 + b_2_4·b_2_5·c_4_16 + b_2_42·c_4_16 + c_4_16·a_1_0·a_3_3
       + b_2_4·c_4_16·a_1_12
  41. b_4_13·b_4_14 + b_4_132 + b_2_52·b_4_14 + b_2_52·b_4_13 + b_2_53·b_1_22 + b_2_54
       + b_2_42·b_4_14 + b_2_42·b_4_13 + b_2_42·b_2_5·b_1_22 + b_2_42·b_2_52
       + b_4_13·b_1_2·a_3_3 + b_2_52·b_1_2·a_3_5 + b_2_52·b_1_2·a_3_3
       + b_2_4·b_2_5·b_1_2·a_3_5 + a_3_5·a_5_16 + b_2_42·a_1_1·a_3_5 + b_2_43·a_1_12
       + b_2_4·a_1_12·b_1_2·a_3_5 + b_2_5·c_4_16·b_1_22 + b_2_52·c_4_16
       + b_2_4·c_4_16·b_1_22 + b_2_4·b_2_5·c_4_16 + b_2_42·c_4_16 + c_4_16·a_1_0·a_3_3
       + c_4_15·a_1_1·a_3_5 + c_4_15·a_1_0·a_3_3 + b_2_4·c_4_16·a_1_12
  42. b_4_13·a_5_16 + b_2_5·b_4_13·a_3_3 + b_2_53·a_3_5 + b_2_53·a_3_3 + b_2_54·a_1_0
       + b_2_4·b_2_5·a_5_16 + b_2_4·b_2_52·a_3_5 + b_2_4·b_2_52·a_3_3 + b_2_42·a_5_16
       + b_2_43·a_1_12·b_1_2 + b_2_5·c_4_16·a_3_5 + b_2_5·c_4_16·a_3_3 + b_2_5·c_4_16·a_3_7
       + b_2_4·c_4_16·a_3_3 + b_2_4·c_4_15·a_3_7 + b_2_42·c_4_16·a_1_1 + b_2_42·c_4_15·a_1_1
       + b_2_42·c_4_15·a_1_0 + b_2_4·c_4_16·a_1_12·b_1_2 + b_2_4·c_4_15·a_1_12·b_1_2
       + c_4_16·a_1_12·a_3_5
  43. b_4_14·a_5_16 + b_2_52·a_5_16 + b_2_53·a_3_5 + b_2_53·a_3_3 + b_2_4·b_2_5·a_5_16
       + b_2_42·a_1_1·b_1_2·a_3_5 + b_2_5·c_4_16·a_3_3 + c_4_15·a_1_1·b_1_2·a_3_5
  44. a_5_162 + b_2_42·a_1_0·a_5_16 + b_2_44·a_1_12 + b_2_4·c_4_16·a_1_0·a_3_3
       + c_4_152·a_1_12

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_15, a Duflot regular element of degree 4
    2. c_4_16, a Duflot regular element of degree 4
    3. b_2_5 + b_2_4, 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, -1, 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. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 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. a_3_7 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_3_5 → 0, an element of degree 3
  9. b_4_13 → 0, an element of degree 4
  10. b_4_14 → 0, an element of degree 4
  11. c_4_15 → c_1_04, an element of degree 4
  12. c_4_16 → c_1_14 + c_1_04, an element of degree 4
  13. a_5_16 → 0, an element of degree 5

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_2_4 → c_1_22, an element of degree 2
  5. b_2_5 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. a_3_7 → 0, an element of degree 3
  7. a_3_3 → 0, an element of degree 3
  8. a_3_5 → 0, an element of degree 3
  9. b_4_13 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_22
       + c_1_02·c_1_32 + c_1_02·c_1_22, an element of degree 4
  10. b_4_14 → c_1_34 + c_1_22·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_02·c_1_32
       + c_1_02·c_1_2·c_1_3, an element of degree 4
  11. c_4_15 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_22
       + c_1_04, an element of degree 4
  12. c_4_16 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_22
       + c_1_14 + c_1_02·c_1_32 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. a_5_16 → 0, 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