Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41618 |
| 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 | |
4648 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);(4,7)(8,10)(11,12,16,20)(13,21,22,17)(14,18,24,26)(15,19,25,23)(27,28,32,35)(29,36,30,33)(31,34,39,42)(37,41,38,40);(13,22)(15,25)(17,21)(19,23)(27,29,32,30)(28,33,35,36)(31,38,39,37)(34,41,42,40);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,19)(15,18)(16,22)(20,21)(23,24)(25,26)(27,30,32,29)(28,33,35,36)(31,38,39,37)(34,40,42,41);(11,14)(12,18)(13,23)(15,21)(16,24)(17,25)(19,22)(20,26)(27,28)(29,33)(30,36)(31,34)(32,35)(37,40)(38,41)(39,42);(3,5,6,9)(4,8,7,10)(11,15,16,25)(12,19,20,23)(13,18,22,26)(14,21,24,17)(27,31,32,39)(28,34,35,42)(29,37,30,38)(33,40,36,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546898);