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×#56903 |
| 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);(3,4,7,5)(6,9,10,8)(11,12)(13,17)(14,18)(15,16)(19,21)(20,25)(22,24)(23,26)(27,28)(29,30)(31,39)(32,36)(33,34)(35,40)(37,41)(38,42);(4,5)(8,9)(12,15)(13,19)(14,22)(25,26)(27,28,32,36)(29,33,34,30)(31,39,40,35)(37,38,42,41);(11,13,16,19)(12,17,15,21)(14,23,22,20)(18,26,24,25)(27,29)(28,33)(30,36)(31,37)(32,34)(35,41)(38,39)(40,42);(3,5,7,4)(6,9,10,8)(11,14,16,22)(12,18,15,24)(13,20,19,23)(17,25,21,26)(27,30,32,33)(28,34,36,29)(31,38,40,41)(35,42,39,37);(3,6,7,10)(4,8,5,9)(11,15,16,12)(13,21,19,17)(14,24,22,18)(20,26,23,25)(27,31)(28,35)(29,37)(30,38)(32,40)(33,41)(34,42)(36,39)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006887);