Cohomology of group number 479 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 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  +  t3  −  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 11 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_1_2, a nilpotent element of degree 1
  4. b_2_3, an element of degree 2
  5. b_2_4, an element of degree 2
  6. a_4_3, a nilpotent element of degree 4
  7. b_5_7, an element of degree 5
  8. b_5_8, an element of degree 5
  9. b_5_9, an element of degree 5
  10. b_8_15, an element of degree 8
  11. c_8_17, a Duflot regular element of degree 8

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Ring relations

There are 30 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_12
  3. a_1_22 + a_1_0·a_1_1
  4. b_2_4·a_1_0 + b_2_3·a_1_1
  5. b_2_3·b_2_4·a_1_1 + b_2_32·a_1_1
  6. a_4_3·a_1_1
  7. a_4_3·a_1_0
  8. a_1_0·b_5_7 + b_2_3·a_4_3
  9. a_1_1·b_5_8 + a_1_1·b_5_7
  10. a_1_1·b_5_7 + a_1_0·b_5_8
  11. a_1_1·b_5_9 + a_1_1·b_5_7 + b_2_4·a_4_3
  12. a_1_0·b_5_9
  13. b_2_4·b_5_7 + b_2_3·b_5_9 + b_2_3·b_5_8 + b_2_33·a_1_1
  14. a_4_32
  15. a_4_3·b_5_7
  16. a_4_3·b_5_8
  17. a_4_3·b_5_9
  18. b_8_15·a_1_1 + b_2_34·a_1_1
  19. b_8_15·a_1_0 + b_2_34·a_1_1
  20. b_5_72 + b_2_33·b_2_42 + b_2_34·b_2_4
  21. b_5_82 + b_2_32·b_2_43 + b_2_33·b_2_42
  22. b_5_92 + b_2_3·b_2_44 + b_2_33·b_2_42
  23. b_5_7·b_5_9 + b_5_7·b_5_8 + b_2_32·b_2_43 + b_2_33·b_2_42 + b_2_32·b_2_4·a_4_3
       + c_8_17·a_1_0·a_1_1
  24. b_5_7·b_5_9 + b_2_3·b_8_15 + b_2_34·b_2_4 + b_2_32·b_2_4·a_4_3
  25. b_5_8·b_5_9 + b_5_7·b_5_9 + b_5_7·b_5_8 + b_2_4·b_8_15 + b_2_3·b_2_44 + b_2_32·b_2_43
       + b_2_33·b_2_42 + b_2_43·a_4_3 + b_2_32·b_2_4·a_4_3
  26. a_4_3·b_8_15 + b_2_33·b_2_4·a_4_3
  27. b_8_15·b_5_8 + b_2_3·b_2_43·b_5_8 + b_2_32·b_2_42·b_5_9 + b_2_33·b_2_4·b_5_9
  28. b_8_15·b_5_9 + b_2_3·b_2_43·b_5_9 + b_2_3·b_2_43·b_5_8 + b_2_33·b_2_4·b_5_8
  29. b_8_15·b_5_7 + b_2_32·b_2_42·b_5_9 + b_2_33·b_2_4·b_5_9 + b_2_34·b_5_9
       + b_2_34·b_5_8 + b_2_36·a_1_1
  30. b_8_152 + b_2_32·b_2_46 + b_2_33·b_2_45 + b_2_34·b_2_44 + b_2_35·b_2_43
       + b_2_36·b_2_42


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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_17, a Duflot regular element of degree 8
    2. 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, 3, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].


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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. a_1_20, an element of degree 1
  4. b_2_30, an element of degree 2
  5. b_2_40, an element of degree 2
  6. a_4_30, an element of degree 4
  7. b_5_70, an element of degree 5
  8. b_5_80, an element of degree 5
  9. b_5_90, an element of degree 5
  10. b_8_150, an element of degree 8
  11. c_8_17c_1_08, an element of degree 8

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

  1. a_1_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. a_1_20, an element of degree 1
  4. b_2_3c_1_12, an element of degree 2
  5. b_2_4c_1_22, an element of degree 2
  6. a_4_30, an element of degree 4
  7. b_5_7c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  8. b_5_8c_1_12·c_1_23 + c_1_13·c_1_22, an element of degree 5
  9. b_5_9c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  10. b_8_15c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
       + c_1_16·c_1_22, an element of degree 8
  11. c_8_17c_1_12·c_1_26 + 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