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×#54343 |
| 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 | |
4452 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)(5,9)(6,8)(7,10)(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,32,36)(29,33,37,42)(30,34,40,39)(31,35,41,38);(3,5,8,7)(4,9,6,10)(12,16)(13,19)(14,23)(24,26)(29,37)(31,41)(33,42)(35,38);(3,6,8,4)(5,9,7,10)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29)(28,33)(30,35)(31,39)(32,37)(34,41)(36,42)(38,40);(3,7,8,5)(4,9,6,10)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,26)(23,25)(27,30)(28,34)(29,38)(31,33)(32,40)(35,37)(36,39)(41,42);(11,14,15,23)(12,18,16,22)(13,21,19,25)(17,24,20,26)(27,31)(28,35)(29,39)(30,33)(32,41)(34,37)(36,38)(40,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010969);