Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13236 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1150 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);(4,6)(7,8,9,10)(11,12)(13,24)(14,30)(15,33)(16,34)(17,23)(18,26)(19,27)(20,32)(21,25)(22,31)(28,42)(29,35)(36,39)(37,41)(38,40);(3,4,5,6)(8,10)(11,13,25,39,16,28,14,26,17,29,32,41,31,40,15,27)(12,18,33,42,22,36,20,24,23,19,30,38,34,37,21,35);(11,13,25,39,16,28,14,26,17,29,32,41,31,40,15,27)(12,19,33,38,22,37,20,35,23,18,30,42,34,36,21,24)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402402);