Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12916 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | #120 |
| commutator | C2×C4 | C22×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1850 |
| prime radical | C2×#41×C127 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G | |
370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,6,12,7,11,26,37,8,13,27,38,5,22,25,39)(3,14,18,41,16,9,28,31,17,32,47,57,23,30,50,56)(4,10,20,34,19,35,48,59,21,36,49,60,24,33,51,58)(15,42,45,63,46,54,52,64,44,40,61,62,43,29,53,55)(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);(1,3,6,18,7,16,26,28,8,17,27,47,5,23,25,50)(2,9,12,31,11,32,37,57,13,30,38,56,22,14,39,41)(4,15,20,45,19,46,48,52,21,44,49,61,24,43,51,53)(10,29,34,55,35,42,59,63,36,54,60,64,33,40,58,62);(1,4)(2,10)(3,15)(5,24)(6,20)(7,19)(8,21)(9,29)(11,35)(12,34)(13,36)(14,40)(16,46)(17,44)(18,45)(22,33)(23,43)(25,51)(26,48)(27,49)(28,52)(30,54)(31,55)(32,42)(37,59)(38,60)(39,58)(41,62)(47,61)(50,53)(56,64)(57,63)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5904);