Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13366 |
| 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 | |
1000 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);(7,8,10,12)(9,13,14,11)(15,16,18,21)(17,25,26,43)(19,31,29,44)(20,34,30,24)(22,38,36,27)(23,40,37,33)(28,45,35,41)(32,42,46,39);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(16,21)(22,36)(23,37)(24,34)(25,43)(31,44)(39,42)(41,45);(4,6)(15,17,18,26)(16,22,21,36)(19,32,29,46)(20,35,30,28)(23,41,37,45)(24,42,34,39)(25,44,43,31)(27,40,38,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406144);