Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#12902 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | #127 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1843 |
| prime radical | C23×C127 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G | |
278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,5,10,6,11,21,32,7,12,22,33,23,34,47,58)(3,8,15,26,16,27,40,52,14,25,38,50,39,51,60,63)(4,9,19,30,20,31,46,57,18,29,44,55,45,56,62,64)(13,35,37,59,17,41,43,61,36,24,48,49,42,28,53,54)(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)(2,8)(4,17,18,42)(5,15)(6,16)(7,14)(9,28,29,41)(10,26)(11,27)(12,25)(13,20,36,45)(19,43,44,53)(21,40)(22,38)(23,39)(24,31,35,56)(30,54,55,61)(32,52)(33,50)(34,51)(37,46,48,62)(47,60)(49,57,59,64)(58,63);(1,4)(2,9)(3,13,14,36)(5,19)(6,20)(7,18)(8,24,25,35)(10,30)(11,31)(12,29)(15,37,38,48)(16,17,39,42)(21,46)(22,44)(23,45)(26,49,50,59)(27,28,51,41)(32,57)(33,55)(34,56)(40,43,60,53)(47,62)(52,54,63,61)(58,64)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5890);