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×#54567 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4750 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);(3,4,8,6)(5,9,7,10)(11,12)(13,18)(14,19)(15,23)(16,17)(20,24)(21,22)(25,26)(27,28,32,35)(29,30,37,33)(31,39,40,34)(36,41,42,38);(4,6)(5,7)(12,17)(14,18)(20,23)(22,25)(27,29,32,37)(28,33,35,30)(31,36,40,42)(34,41,39,38);(3,5,8,7)(4,9,6,10)(11,13,16,19)(12,18,17,14)(15,21,24,26)(20,25,23,22)(27,29)(28,33)(30,35)(31,36)(32,37)(34,41)(38,39)(40,42);(3,6,8,4)(5,9,7,10)(11,14,16,18)(12,19,17,13)(15,22,24,25)(20,26,23,21)(27,30,32,33)(28,29,35,37)(31,38,40,41)(34,36,39,42);(3,7,8,5)(4,9,6,10)(11,15,16,24)(12,20,17,23)(13,21,19,26)(14,22,18,25)(27,31)(28,34)(29,36)(30,38)(32,40)(33,41)(35,39)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010877);