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×#56067 |
| 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 | |
4550 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);(3,4,6,7)(5,8,9,10)(11,12,15,18)(13,19,20,16)(14,17,22,24)(21,25,26,23)(27,28)(29,36)(30,33)(31,39)(32,35)(34,40)(37,41)(38,42);(4,7)(5,9)(13,20)(16,19)(21,26)(23,25)(27,29,32,30)(28,33,35,36)(31,37,40,38)(34,41,39,42);(3,5,6,9)(4,8,7,10)(11,13,15,20)(12,16,18,19)(14,21,22,26)(17,23,24,25)(27,30,32,29)(28,33,35,36)(31,38,40,37)(34,41,39,42);(3,5)(4,8)(6,9)(7,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,36)(30,33)(31,34)(32,35)(37,42)(38,41)(39,40);(3,4)(5,8)(6,7)(9,10)(11,14,15,22)(12,17,18,24)(13,21,20,26)(16,23,19,25)(27,31)(28,34)(29,37)(30,38)(32,40)(33,41)(35,39)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010326);