Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7657 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,7,8,9)(11,12)(13,19,21,16)(14,17)(15,18)(20,25,26,23)(22,24)(27,28,30,33,31,34,37,40)(29,32,36,39,35,38,41,42);(3,4)(5,9)(6,8)(7,10)(11,13,15,21)(12,16,18,19)(14,20,22,26)(17,23,24,25)(27,29)(28,32)(30,36)(31,35)(33,39)(34,38)(37,41)(40,42);(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)(29,35)(32,38)(36,41)(39,42)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68525);