Cohomology of group number 593 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 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.


Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1) · (t4  +  t2  +  t  +  1)

    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-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. b_1_2, an element of degree 1
  4. b_2_3, an element of degree 2
  5. b_2_4, an element of degree 2
  6. b_2_5, an element of degree 2
  7. a_4_6, a nilpotent element of degree 4
  8. b_5_15, an element of degree 5
  9. b_5_16, an element of degree 5
  10. a_8_16, a nilpotent element of degree 8
  11. c_8_29, 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_12 + a_1_02
  2. a_1_0·a_1_1
  3. a_1_0·b_1_2
  4. b_2_3·a_1_0
  5. b_2_5·b_1_2 + b_2_4·b_1_2 + b_2_5·a_1_0 + b_2_4·a_1_1
  6. b_2_4·b_1_22 + b_2_32 + b_2_3·a_1_1·b_1_2 + b_2_4·a_1_02
  7. b_2_4·b_2_5·a_1_1 + b_2_4·b_2_5·a_1_0 + b_2_32·a_1_1
  8. a_4_6·b_1_2 + b_2_3·a_1_1·b_1_22 + b_2_3·b_2_5·a_1_1
  9. a_4_6·a_1_1
  10. b_2_3·b_2_5·a_1_1 + b_2_3·b_2_4·a_1_1 + a_4_6·a_1_0
  11. b_2_3·a_4_6
  12. a_1_1·b_5_15 + a_1_0·b_5_15 + b_2_3·b_2_4·a_1_1·b_1_2 + b_2_42·a_1_02
  13. b_1_2·b_5_16 + b_1_2·b_5_15 + b_2_3·b_2_4·b_2_5 + b_2_3·b_2_42 + b_2_32·b_2_4
       + b_2_33 + b_2_3·a_1_1·b_1_23 + b_2_42·a_1_02
  14. b_2_3·b_2_4·b_2_5 + b_2_3·b_2_42 + a_1_1·b_5_16 + a_1_1·b_5_15 + b_2_32·a_1_1·b_1_2
       + b_2_42·a_1_02
  15. a_1_0·b_5_16 + b_2_4·a_4_6
  16. b_2_3·b_5_16 + b_2_3·b_5_15 + b_2_3·b_2_42·b_1_2 + b_2_32·b_2_4·b_1_2
       + b_2_4·b_2_52·a_1_0 + b_2_42·b_2_5·a_1_0 + b_2_32·b_2_4·a_1_1 + b_2_33·a_1_1
       + b_2_5·a_4_6·a_1_0 + b_2_4·a_4_6·a_1_0
  17. a_4_62
  18. b_2_3·b_2_5·b_5_15 + b_2_3·b_2_4·b_5_15 + a_4_6·b_5_15 + b_2_4·b_2_53·a_1_0
       + b_2_34·a_1_1 + b_2_42·a_4_6·a_1_0
  19. a_4_6·b_5_16 + b_2_43·b_2_5·a_1_0 + b_2_42·a_4_6·a_1_0
  20. b_2_3·b_1_22·b_5_15 + b_2_3·b_2_5·b_5_15 + b_2_3·b_2_4·b_5_15 + b_2_3·b_2_43·b_1_2
       + b_2_32·b_5_15 + b_2_32·b_2_42·b_1_2 + a_8_16·b_1_2 + b_2_4·b_2_53·a_1_0
       + b_2_42·b_2_52·a_1_0 + b_2_34·a_1_1 + b_2_52·a_4_6·a_1_0 + b_2_42·a_4_6·a_1_0
  21. b_2_3·b_2_5·b_5_15 + b_2_3·b_2_4·b_5_15 + b_2_4·b_2_53·a_1_0 + b_2_42·b_2_52·a_1_0
       + b_2_34·a_1_1 + a_8_16·a_1_1 + b_2_52·a_4_6·a_1_0
  22. b_2_3·b_2_5·b_5_15 + b_2_3·b_2_4·b_5_15 + b_2_4·b_2_53·a_1_0 + b_2_42·b_2_52·a_1_0
       + b_2_34·a_1_1 + a_8_16·a_1_0 + b_2_52·a_4_6·a_1_0
  23. b_5_152 + b_2_42·b_2_53 + b_2_33·b_2_4·a_1_1·b_1_2 + b_2_34·a_1_1·b_1_2
       + b_2_44·a_1_02
  24. b_5_162 + b_2_44·b_2_5 + b_2_32·b_2_43 + b_2_34·b_2_4 + b_2_42·a_1_1·b_5_16
       + b_2_43·a_4_6 + b_2_33·b_2_4·a_1_1·b_1_2 + c_8_29·a_1_02
  25. b_2_3·b_2_4·b_1_2·b_5_15 + b_2_32·b_1_2·b_5_15 + b_2_32·b_2_43 + b_2_33·b_2_42
       + b_2_4·b_2_5·a_1_0·b_5_15 + b_2_4·b_2_52·a_4_6 + b_2_3·a_8_16
       + b_2_33·b_2_4·a_1_1·b_1_2
  26. b_5_162 + b_5_15·b_5_16 + b_2_43·b_2_52 + b_2_44·b_2_5 + b_2_3·b_2_44
       + b_2_34·b_2_4 + b_2_4·a_8_16 + b_2_4·b_2_5·a_1_0·b_5_15 + b_2_42·b_2_5·a_4_6
       + b_2_33·b_2_4·a_1_1·b_1_2 + b_2_34·a_1_1·b_1_2 + b_2_44·a_1_02
  27. b_2_42·b_2_5·a_1_0·b_5_15 + b_2_42·b_2_52·a_4_6 + a_4_6·a_8_16
  28. b_2_4·b_2_53·b_5_16 + b_2_42·b_2_52·b_5_15 + b_2_3·b_2_43·b_5_15
       + b_2_32·b_2_42·b_5_15 + a_8_16·b_5_15 + b_2_42·b_2_54·a_1_0
       + b_2_43·b_2_53·a_1_0 + b_2_35·b_2_4·a_1_1 + b_2_36·a_1_1
       + b_2_4·b_2_5·a_8_16·a_1_0
  29. b_2_4·b_2_53·b_5_16 + b_2_42·b_2_52·b_5_16 + b_2_42·b_2_52·b_5_15
       + b_2_43·b_2_5·b_5_15 + a_8_16·b_5_16 + a_8_16·b_5_15 + b_2_42·a_8_16·b_1_2
       + b_2_42·b_2_54·a_1_0 + b_2_45·b_2_5·a_1_0 + b_2_3·b_2_4·a_8_16·b_1_2
       + b_2_35·b_2_4·a_1_1 + b_2_36·a_1_1 + b_2_54·a_4_6·a_1_0 + b_2_42·a_8_16·a_1_0
       + a_4_6·c_8_29·a_1_0
  30. a_8_162


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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_29, a Duflot regular element of degree 8
    2. b_1_24 + b_2_52 + b_2_4·b_2_5 + b_2_42 + b_2_32, an element of degree 4
    3. b_1_2·b_5_15 + b_2_4·b_2_52 + b_2_42·b_2_5 + b_2_3·b_2_42 + b_2_32·b_1_22
         + b_2_32·b_2_4, an element of degree 6
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 15].
  • 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. b_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. b_2_50, an element of degree 2
  7. a_4_60, an element of degree 4
  8. b_5_150, an element of degree 5
  9. b_5_160, an element of degree 5
  10. a_8_160, an element of degree 8
  11. c_8_29c_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. b_1_2c_1_1, an element of degree 1
  4. b_2_3c_1_1·c_1_2, an element of degree 2
  5. b_2_4c_1_22, an element of degree 2
  6. b_2_5c_1_22, an element of degree 2
  7. a_4_60, an element of degree 4
  8. b_5_15c_1_25, an element of degree 5
  9. b_5_16c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
  10. a_8_160, an element of degree 8
  11. c_8_29c_1_1·c_1_27 + c_1_13·c_1_25 + 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

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. b_1_20, an element of degree 1
  4. b_2_30, an element of degree 2
  5. b_2_4c_1_22 + c_1_12, an element of degree 2
  6. b_2_5c_1_22, an element of degree 2
  7. a_4_60, an element of degree 4
  8. b_5_15c_1_25 + c_1_12·c_1_23, an element of degree 5
  9. b_5_16c_1_25 + c_1_14·c_1_2, an element of degree 5
  10. a_8_160, an element of degree 8
  11. c_8_29c_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


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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