Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13339 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 195 points:
(5,6,12,18)(7,19,24,54)(8,25,31,44)(9,32,37,50)(10,17,11,16)(13,38,42,21)(14,27,47,28)(15,35,51,36)(20,39,58,67)(22,48,23,53)(26,63,57,43)(29,49,30,62)(33,40,60,41)(34,45,65,46)(52,55,68,56)(59,61,64,66)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(1,2)(3,4)(5,7,12,24)(6,13,18,42)(8,26,31,57)(9,33,37,60)(10,23,11,22)(14,43,47,63)(15,48,51,53)(16,41,17,40)(19,50,54,32)(20,30,58,29)(21,35,38,36)(25,61,44,66)(27,55,28,56)(34,64,65,59)(39,46,67,45)(49,68,62,52);(2,4)(5,8)(6,14)(7,20)(9,34)(10,30)(11,29)(12,31)(13,39)(15,49)(16,46)(17,45)(18,47)(19,52)(21,59)(22,57)(23,26)(24,58)(25,32)(27,35)(28,36)(33,56)(37,65)(38,64)(40,63)(41,43)(42,67)(44,50)(48,61)(51,62)(53,66)(54,68)(55,60)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6327);