Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#13641 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C23 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C23 | C22×C16 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 74 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 74 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,14,19,15,20,27,37,16,21,28,38,30,40,49,59,17,22,29,39,31,41,50,60,32,42,51,61,52,62,67,72)(13,18,24,34,25,35,46,56,26,36,47,57,48,58,66,71,23,33,43,53,44,54,63,68,45,55,64,69,65,70,73,74);(4,7)(8,10)(11,13)(12,18)(14,24)(15,25)(16,26)(17,23)(19,34)(20,35)(21,36)(22,33)(27,46)(28,47)(29,43)(30,48)(31,44)(32,45)(37,56)(38,57)(39,53)(40,58)(41,54)(42,55)(49,66)(50,63)(51,64)(52,65)(59,71)(60,68)(61,69)(62,70)(67,73)(72,74);(3,5,6,9)(4,8,7,10)(13,23)(18,33)(24,43)(25,44)(26,45)(34,53)(35,54)(36,55)(46,63)(47,64)(48,65)(56,68)(57,69)(58,70)(66,73)(71,74)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419923);