Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13483 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
804 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)(9,12)(10,11)(13,14)(15,16)(17,21)(18,22)(19,23)(20,24)(25,39)(26,34)(27,35)(28,36)(29,37)(30,38)(31,41)(32,42)(33,43)(40,45)(44,46);(8,11)(12,14)(15,17,18,26)(16,21,22,34)(19,25,29,43)(20,32,30,28)(23,39,37,33)(24,42,38,36)(27,45,44,31)(35,40,46,41);(4,6)(7,9,10,13)(8,12,11,14)(16,22)(17,25)(19,31)(20,30)(21,33)(23,40)(26,43)(27,28)(29,45)(32,44)(34,39)(35,42)(36,46)(37,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406540);