Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57618 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,19)(14,22)(15,16)(17,21)(18,23)(20,25)(24,26)(27,28,32,35)(29,30,37,33)(31,39,40,34)(36,41,42,38);(6,10)(8,9)(12,15)(14,23)(17,19)(20,24)(27,29,32,37)(28,33,35,30)(31,36,40,42)(34,41,39,38);(4,5)(8,9)(11,13)(12,17)(14,24)(15,19)(16,21)(18,26)(20,23)(22,25)(27,30,32,33)(28,29,35,37)(31,38,40,41)(34,36,39,42);(3,5,7,4)(6,9,10,8)(11,14)(12,18)(13,20)(15,22)(16,23)(17,25)(19,26)(21,24)(27,29)(28,33)(30,35)(31,36)(32,37)(34,41)(38,39)(40,42);(3,6)(4,8)(5,9)(7,10)(11,15,16,12)(13,19,21,17)(14,22,23,18)(20,26,24,25)(27,31,32,40)(28,34,35,39)(29,36,37,42)(30,38,33,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009008);