Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C127×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,8,14)(3,15,20,48)(4,21,25,45)(5,26,31,43)(6,13,7,12)(9,33,37,17)(10,32,40,24)(11,29,44,30)(16,49,52,64)(18,47,19,41)(22,56,51,46)(23,42,28,39)(27,36,54,35)(34,60,63,57)(38,53,62,50)(55,61,59,58)(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)(2,9)(4,22)(5,27)(6,18)(7,19)(8,20)(10,38)(11,41)(12,35)(13,36)(14,37)(15,26)(16,24)(17,30)(21,34)(23,53)(25,51)(28,50)(29,33)(31,54)(32,52)(39,46)(40,62)(42,56)(43,48)(44,47)(45,63)(49,55)(57,58)(59,64)(60,61);(1,4)(2,10)(3,16)(5,28)(6,32)(7,24)(8,25)(9,34)(11,42)(12,45)(13,21)(14,40)(15,46)(17,53)(18,22)(19,51)(20,52)(23,31)(26,59)(27,60)(29,58)(30,61)(33,50)(35,38)(36,62)(37,63)(39,44)(41,64)(43,55)(47,49)(48,56)(54,57)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6312);