Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13525 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #2092 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,5,11,30,46,6,12,8,14,32,48,7,13,31,47)(3,15,17,37,19,36,18,39,21,49,33,38,20,44,28,9)(4,22,24,10,34,41,29,50,27,45,59,43,26,57,25,42)(16,40,52,51,23,58,56,64,55,63,60,62,54,35,53,61)(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);(1,3,8,21)(2,9,14,39)(4,23)(5,28,32,18)(6,33,31,17)(7,19,30,20)(10,40)(11,44,48,36)(12,49,47,15)(13,37,46,38)(16,34)(22,51)(24,52)(25,56)(26,55)(27,54)(29,53)(35,50)(41,61)(42,58)(43,63)(45,62)(57,64)(59,60);(1,4)(2,10)(3,16)(5,29)(6,24)(7,34)(8,27)(9,35)(11,45)(12,41)(13,50)(14,43)(15,51)(17,56)(18,52)(19,54)(20,23)(21,55)(22,48)(25,32)(26,30)(28,60)(31,59)(33,53)(36,61)(37,63)(38,40)(39,58)(42,46)(44,64)(47,57)(49,62)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6513);