Cohomology of group number 248 of order 128

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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 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 4, respectively.

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. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. a_2_5, a nilpotent element of degree 2
  5. b_2_4, an element of degree 2
  6. a_3_8, a nilpotent element of degree 3
  7. b_3_7, an element of degree 3
  8. b_3_9, an 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. b_5_25, an element of degree 5

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Ring relations

There are 44 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_1·b_1_22 + a_1_0·b_1_22
  4. a_2_5·a_1_0
  5. a_2_5·b_1_1
  6. b_2_4·b_1_1 + a_1_0·b_1_22
  7. a_1_0·b_1_23 + b_2_4·a_1_0·b_1_2 + a_2_52
  8. a_1_0·a_3_8
  9. b_1_1·a_3_8
  10. a_1_0·b_3_7 + a_1_0·b_1_23
  11. b_1_1·b_3_7 + b_1_13·b_1_2 + a_1_0·b_1_23
  12. a_1_0·b_3_9
  13. a_2_5·b_3_9
  14. b_1_22·b_3_9 + b_1_22·a_3_8 + b_2_4·a_1_0·b_1_22 + a_2_5·b_3_7 + a_2_5·b_1_23
  15. b_2_4·b_3_9 + b_1_22·a_3_8 + b_2_4·a_1_0·b_1_22 + a_2_5·b_3_7 + a_2_5·b_1_23
       + a_2_5·a_3_8
  16. b_1_22·a_3_8 + b_4_13·a_1_0 + a_2_5·b_3_7 + a_2_5·b_1_23
  17. b_1_1·b_1_2·b_3_9 + b_4_13·b_1_1 + b_1_22·a_3_8 + a_2_5·b_3_7 + a_2_5·b_1_23
  18. b_4_14·a_1_0 + a_2_5·a_3_8
  19. b_1_1·b_1_2·b_3_9 + b_4_14·b_1_1
  20. a_3_82 + a_2_52·b_2_4
  21. b_3_72 + b_2_4·b_1_24
  22. a_3_8·b_3_9
  23. b_3_92 + c_4_15·b_1_12
  24. a_3_8·b_3_7 + b_4_13·a_1_0·b_1_2 + a_2_5·b_1_2·b_3_7 + a_2_5·b_2_4·b_1_22
       + a_2_5·b_1_2·a_3_8
  25. b_3_7·b_3_9 + b_4_13·b_1_12 + a_3_8·b_3_7 + b_2_42·a_1_0·b_1_2 + a_2_5·b_1_2·b_3_7
       + a_2_5·b_2_4·b_1_22 + a_2_5·b_1_2·a_3_8 + a_2_52·b_2_4
  26. a_3_8·b_3_7 + b_2_4·b_1_2·a_3_8 + a_2_5·b_4_14 + a_2_52·b_2_4
  27. b_1_23·b_3_7 + b_4_14·b_1_22 + b_2_4·b_1_2·b_3_7 + a_3_8·b_3_7 + b_2_42·a_1_0·b_1_2
       + a_2_5·b_1_24 + a_2_5·b_4_13 + a_2_5·b_2_4·b_1_22 + a_2_52·b_2_4
  28. a_1_0·b_5_25 + b_2_42·a_1_0·b_1_2 + a_2_52·b_2_4 + c_4_16·a_1_0·b_1_2
  29. b_1_1·b_5_25 + b_2_42·a_1_0·b_1_2 + a_2_52·b_2_4 + c_4_16·b_1_1·b_1_2
       + c_4_16·b_1_12
  30. b_4_13·b_3_9 + b_2_42·a_1_0·b_1_22 + c_4_15·b_1_12·b_1_2 + c_4_15·a_1_0·b_1_22
  31. b_4_14·b_3_7 + b_2_4·b_1_25 + b_2_42·b_1_23 + b_4_14·a_3_8 + b_4_13·a_3_8
       + a_2_5·b_4_13·b_1_2 + a_2_5·b_2_4·b_3_7 + a_2_5·b_2_42·b_1_2 + a_2_5·b_2_4·a_3_8
       + c_4_15·a_1_0·b_1_22
  32. b_4_14·a_3_8 + a_2_5·b_4_14·b_1_2 + a_2_5·b_2_4·b_1_23 + a_2_5·b_2_42·b_1_2
       + a_2_5·b_2_4·a_3_8
  33. b_4_14·b_3_9 + c_4_15·b_1_12·b_1_2
  34. b_4_13·a_3_8 + b_2_4·b_4_13·a_1_0 + b_2_42·a_1_0·b_1_22 + a_2_5·b_5_25
       + a_2_5·b_2_4·b_3_7 + a_2_5·b_2_4·a_3_8 + c_4_15·a_1_0·b_1_22 + a_2_5·c_4_16·b_1_2
  35. b_1_22·b_5_25 + b_4_13·b_3_7 + b_4_13·b_1_23 + b_2_4·b_1_22·b_3_7
       + b_2_42·a_1_0·b_1_22 + a_2_5·b_1_25 + a_2_5·b_4_13·b_1_2 + a_2_5·b_2_4·b_3_7
       + a_2_5·b_2_4·b_1_23 + c_4_16·b_1_23 + c_4_16·a_1_0·b_1_22
  36. b_4_142 + b_2_4·b_1_26 + b_2_43·b_1_22 + a_2_52·b_2_42 + a_2_52·c_4_15
  37. b_4_13·b_1_24 + b_4_132 + b_2_4·b_4_14·b_1_22 + b_2_42·b_1_2·b_3_7
       + b_2_42·b_1_2·a_3_8 + b_2_43·a_1_0·b_1_2 + a_2_5·b_1_26 + a_2_5·b_4_14·b_1_22
       + a_2_5·b_2_4·b_1_2·b_3_7 + a_2_5·b_2_4·b_4_14 + a_2_5·b_2_4·b_4_13
       + a_2_5·b_2_42·b_1_22 + c_4_15·b_1_24 + a_2_52·c_4_16
  38. a_3_8·b_5_25 + b_2_42·b_1_2·a_3_8 + a_2_5·b_4_13·b_1_22 + a_2_5·b_2_4·b_4_14
       + a_2_5·b_2_4·b_4_13 + c_4_16·b_1_2·a_3_8
  39. b_3_7·b_5_25 + b_4_13·b_1_24 + b_4_13·b_4_14 + b_4_132 + b_2_4·b_1_2·b_5_25
       + b_2_4·b_4_14·b_1_22 + b_2_42·b_1_24 + b_2_42·b_1_2·a_3_8 + a_2_5·b_1_26
       + a_2_5·b_2_4·b_1_2·b_3_7 + a_2_5·b_2_4·b_1_24 + a_2_5·b_2_42·b_1_22
       + c_4_16·b_1_2·b_3_7 + c_4_16·b_1_13·b_1_2 + c_4_15·b_1_24 + b_2_4·c_4_16·b_1_22
       + b_2_4·c_4_15·a_1_0·b_1_2 + a_2_5·c_4_15·b_1_22 + a_2_52·c_4_16 + a_2_52·c_4_15
  40. b_3_7·b_5_25 + b_4_13·b_1_2·b_3_7 + b_4_13·b_1_24 + b_4_132 + b_2_4·b_4_14·b_1_22
       + b_2_4·b_4_13·b_1_22 + b_2_42·b_1_2·b_3_7 + b_2_42·b_1_24 + b_2_42·b_1_2·a_3_8
       + a_2_5·b_1_2·b_5_25 + a_2_5·b_1_26 + a_2_5·b_4_13·b_1_22 + a_2_5·b_2_4·b_4_14
       + a_2_5·b_2_4·b_4_13 + a_2_52·b_2_42 + c_4_16·b_1_2·b_3_7 + c_4_16·b_1_13·b_1_2
       + c_4_15·b_1_24 + b_2_4·c_4_16·a_1_0·b_1_2 + a_2_5·c_4_16·b_1_22
  41. b_3_9·b_5_25 + b_2_42·b_1_2·a_3_8 + b_2_43·a_1_0·b_1_2 + a_2_5·b_2_4·b_1_2·b_3_7
       + a_2_5·b_2_4·b_4_14 + a_2_5·b_2_42·b_1_22 + a_2_52·b_4_14 + c_4_16·b_1_2·b_3_9
       + c_4_16·b_1_1·b_3_9
  42. b_4_14·b_5_25 + b_4_13·b_4_14·b_1_2 + b_2_4·b_4_13·b_1_23 + b_2_42·b_1_25
       + b_2_42·b_4_13·b_1_2 + b_2_43·b_1_23 + b_2_42·b_4_13·a_1_0
       + a_2_5·b_4_14·b_1_23 + a_2_5·b_4_13·b_3_7 + a_2_5·b_2_4·b_4_13·b_1_2
       + a_2_5·b_2_42·b_3_7 + a_2_5·b_2_42·b_1_23 + a_2_5·b_2_43·b_1_2
       + a_2_5·b_2_42·a_3_8 + b_4_14·c_4_16·b_1_2 + b_4_13·c_4_16·b_1_1 + b_4_13·c_4_16·a_1_0
       + b_2_4·c_4_15·a_1_0·b_1_22 + a_2_5·c_4_15·b_3_7
  43. b_4_13·b_5_25 + b_4_13·b_4_14·b_1_2 + b_4_132·b_1_2 + b_2_42·b_1_25
       + b_2_42·b_4_13·a_1_0 + b_2_43·a_1_0·b_1_22 + a_2_5·b_4_14·b_1_23
       + a_2_5·b_4_13·b_3_7 + a_2_5·b_2_4·b_5_25 + a_2_5·b_2_4·b_1_25 + a_2_5·b_2_42·b_3_7
       + a_2_5·b_2_42·a_3_8 + c_4_15·b_1_22·b_3_7 + b_4_13·c_4_16·b_1_2
       + b_4_13·c_4_16·b_1_1 + b_2_4·c_4_15·a_1_0·b_1_22 + a_2_5·b_2_4·c_4_16·b_1_2
       + a_2_5·c_4_16·a_3_8
  44. b_5_252 + b_4_132·b_1_22 + b_2_4·b_4_132 + b_2_43·b_1_24 + a_2_52·b_2_43
       + c_4_162·b_1_22 + c_4_162·b_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_1_24 + b_1_14 + b_2_4·b_1_22 + b_2_42, an element of degree 4
    4. b_3_7 + b_2_4·b_1_2, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms 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 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_5 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. a_3_8 → 0, an element of degree 3
  7. b_3_7 → 0, an element of degree 3
  8. b_3_9 → 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. b_5_25 → 0, an element of degree 5

Restriction map to a maximal el. ab. subgp. of rank 3

  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 → 0, an element of degree 1
  4. a_2_5 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. a_3_8 → 0, an element of degree 3
  7. b_3_7 → 0, an element of degree 3
  8. b_3_9 → c_1_0·c_1_22 + c_1_02·c_1_2, 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_02·c_1_22 + c_1_04, an element of degree 4
  12. c_4_16 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. b_5_25 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, 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. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. a_2_5 → 0, an element of degree 2
  5. b_2_4 → c_1_32, an element of degree 2
  6. a_3_8 → 0, an element of degree 3
  7. b_3_7 → c_1_22·c_1_3, an element of degree 3
  8. b_3_9 → 0, an element of degree 3
  9. b_4_13 → c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  10. b_4_14 → c_1_2·c_1_33 + c_1_23·c_1_3, an element of degree 4
  11. c_4_15 → c_1_2·c_1_33 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  12. c_4_16 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  13. b_5_25 → c_1_22·c_1_33 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_22·c_1_32
       + c_1_0·c_1_24 + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_2, 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