Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13281 |
| 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 | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 46 points:
(1,2);(3,4,5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,27)(18,24)(19,34)(20,36)(21,26)(22,29)(23,32)(25,35)(28,41)(30,37)(31,43)(33,40)(38,45)(39,44)(42,46);(4,6)(8,11)(12,13)(15,17,18,28,20,30,32,44,21,31,33,45,35,46,19,29)(16,22,34,42,25,38,40,43,26,39,23,37,36,41,24,27);(7,9,10,14)(8,12,11,13)(15,18,20,32,21,33,35,19)(16,23,25,24,26,34,36,40)(17,28,30,44,31,45,46,29)(22,37,38,27,39,42,41,43)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402564);