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×#57934 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4680 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);(3,4,8,9)(5,7,6,10)(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,33,35)(29,36,30,31)(32,40,41,34)(37,39,38,42);(4,9)(5,6)(12,16)(13,19)(14,23)(24,26)(27,29,33,30)(28,31,35,36)(32,37,41,38)(34,39,40,42);(3,5,8,6)(4,10,9,7)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,30,33,29)(28,31,35,36)(32,38,41,37)(34,39,40,42);(3,6,8,5)(4,7,9,10)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,26)(23,25)(27,31)(28,29)(30,35)(32,39)(33,36)(34,37)(38,40)(41,42);(3,7)(4,5)(6,9)(8,10)(11,14,15,23)(12,18,16,22)(13,21,19,25)(17,24,20,26)(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, 8008645);