Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13240 |
| 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 | |
1130 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)(14,29)(15,32)(16,34)(17,35)(18,23)(19,26)(20,30)(21,25)(22,33)(27,37)(28,42)(31,39)(36,40)(38,41);(3,4,5,6)(8,10)(11,13,25,40,17,28,15,26,18,14,30,41,33,31,16,27)(12,19,34,42,22,36,20,24,23,37,32,39,35,38,21,29);(11,14,30,40,17,31,16,26,18,13,25,41,33,28,15,27)(12,19,32,39,22,36,21,29,23,37,34,42,35,38,20,24)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402407);