Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13620 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | C2×D16 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C2×C127 | C2×#357 |
| prime radical | C2×#508×C127 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,5,10,7,13,26,12)(3,14,17,44,15,43,19,50)(4,20,23,51,25,53,58,52)(6,27,22,11,29,39,57,41)(8,30,32,45,31,47,42,16)(9,34,36,59,38,61,37,24)(18,49,48,62,46,63,28,35)(21,40,55,60,54,64,56,33)(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)(4,21)(5,17)(6,28)(7,15)(9,35)(10,32)(11,40)(12,42)(13,31)(14,20)(16,24)(18,22)(19,26)(23,55)(25,54)(27,33)(29,48)(30,34)(36,49)(37,63)(38,62)(39,60)(41,64)(43,53)(44,51)(45,59)(46,57)(47,61)(50,52)(56,58);(3,15)(8,31)(14,43)(16,45)(17,19)(18,46)(21,54)(28,48)(30,47)(32,42)(33,60)(35,62)(40,64)(44,50)(49,63)(55,56)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6608);