Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#8490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #379 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C2×C42 | C23×C127 |
| prime radical | #538×C127 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,5,11,27,34,57,60)(3,13,16,42,15,48,25,56)(4,19,22,51,21,37,53,62)(6,9,26,33,7,12,28,40)(8,29,32,14,44,46,39,45)(10,35,38,24,41,55,64,23)(17,43,49,63,18,36,50,61)(20,30,52,58,47,31,54,59)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3,6,17)(2,8,9,30)(4,20,23,45)(5,25,26,50)(7,18,27,15)(10,36,19,48)(11,39,33,59)(12,31,34,44)(13,41,43,37)(14,21,47,24)(16,28,49,57)(22,54,35,46)(29,53,52,55)(32,40,58,60)(38,63,51,42)(56,64,61,62);(2,9)(4,21)(5,26)(8,30)(10,37)(12,34)(13,36)(14,45)(16,49)(19,41)(20,47)(22,55)(23,24)(25,50)(28,57)(29,54)(31,44)(35,53)(38,64)(42,56)(43,48)(46,52)(51,62)(61,63)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 1478);