Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13330 |
| 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 | |
1584 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);(3,4)(5,6)(11,12,17,22)(13,23,14,26)(15,29,34,42)(16,35,32,21)(18,27,19,24)(20,38,41,33)(25,40,28,39)(30,37,31,36);(4,6)(7,8)(9,10)(11,13,17,14)(12,18,22,19)(15,30,34,31)(16,28,32,25)(20,39,41,40)(21,37,35,36)(23,42,26,29)(24,38,27,33);(8,10)(11,14,17,13)(12,19,22,18)(15,31,34,30)(16,25,32,28)(20,40,41,39)(21,36,35,37)(23,29,26,42)(24,33,27,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406039);