Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#19063 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C127 |
| prime radical | #532×C127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 175 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 175 points:
(3,9)(4,11)(6,14)(7,12)(10,13)(15,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,38)(31,39)(33,42)(35,40)(37,41)(43,45)(44,47)(46,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175);(1,2,5,8)(3,6,10,15)(4,12,11,7)(9,16,13,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,29)(27,42)(28,39)(34,47)(35,44)(36,45)(37,46)(40,48)(41,43);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,11)(12,14)(17,20,23,37)(18,25,28,40)(19,30,33,46)(21,36,22,35)(24,38,39,48)(26,41,27,34)(29,43,42,47)(31,45,32,44);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,42)(25,34,40,41)(30,44,46,45)(38,47,48,43)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 12051);