Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C2×C127×#1417 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | D16 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C22×C127 | #357 |
| prime radical | C2×#508×C127 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 193 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 193 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193);(3,4,7,12,9,15,28,14)(5,16,18,45,21,46,20,52)(6,22,25,53,27,55,60,54)(8,29,24,13,31,41,59,43)(10,32,33,48,35,47,44,17)(11,36,38,61,40,63,39,26)(19,50,49,64,51,65,30,37)(23,42,56,62,58,66,57,34);(3,5)(4,10)(6,23)(7,18)(8,30)(9,21)(11,37)(12,33)(13,42)(14,44)(15,35)(16,22)(17,26)(19,24)(20,28)(25,56)(27,58)(29,34)(31,49)(32,36)(38,50)(39,65)(40,64)(41,62)(43,66)(45,53)(46,55)(47,63)(48,61)(51,59)(52,54)(57,60)9652 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6607);