Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#19571 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C127 |
| prime radical | C23×C127 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,10)(3,11,13,14)(4,8,7,16)(5,12,15,9)(18,22)(19,25)(20,28)(24,30)(26,27)(29,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,3,5,8,6,13,15,16)(2,7,9,11,10,4,12,14)(17,18,20,22,21,24,28,30)(19,23,27,29,26,31,25,32);(1,4)(2,8)(3,12)(5,14)(6,7)(9,13)(10,16)(11,15)(17,19,21,26)(18,23,24,31)(20,27,28,25)(22,29,30,32);(3,13)(4,7)(8,16)(11,14)(19,26)(23,31)(25,27)(29,32)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 12559);