Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47432 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4464 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);(5,9)(8,10)(11,12,16,14)(13,19,20,17)(15,18,23,22)(21,25,26,24)(27,28,32,30)(29,35,36,33)(31,38,39,34)(37,40,42,41);(3,4,7,6)(5,8,9,10)(13,20)(15,23)(17,19)(18,22)(28,30)(29,36)(31,39)(40,41);(3,5)(4,8)(6,10)(7,9)(11,13)(12,17)(14,19)(15,21)(16,20)(18,24)(22,25)(23,26)(27,29,32,36)(28,33,30,35)(31,37,39,42)(34,40,38,41);(3,4,7,6)(5,8,9,10)(11,14,16,12)(13,17,20,19)(15,22,23,18)(21,24,26,25)(27,30,32,28)(29,33,36,35)(31,38,39,34)(37,40,42,41);(3,6,7,4)(5,10,9,8)(11,15,16,23)(12,18,14,22)(13,21,20,26)(17,24,19,25)(27,31)(28,34)(29,37)(30,38)(32,39)(33,40)(35,41)(36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608531);