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×#54226 |
| 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 | |
4566 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);(4,6)(5,7)(11,12,16,20)(13,17,21,26)(14,18,24,23)(15,19,25,22)(27,28,33,35)(29,36,30,31)(32,40,41,34)(37,39,38,42);(3,4)(5,9)(6,8)(7,10)(13,21)(15,25)(17,26)(19,22)(27,29,33,30)(28,31,35,36)(32,37,41,38)(34,39,40,42);(3,5,8,7)(4,9,6,10)(11,13)(12,17)(14,19)(15,23)(16,21)(18,25)(20,26)(22,24)(27,30,33,29)(28,31,35,36)(32,38,41,37)(34,39,40,42);(3,6,8,4)(5,9,7,10)(11,14)(12,18)(13,22)(15,17)(16,24)(19,21)(20,23)(25,26)(27,31)(28,29)(30,35)(32,39)(33,36)(34,37)(38,40)(41,42);(3,7,8,5)(4,9,6,10)(11,15)(12,19)(13,23)(14,17)(16,25)(18,21)(20,22)(24,26)(27,32,33,41)(28,34,35,40)(29,37,30,38)(31,39,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010948);