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×#13530 |
| 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 | |
892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(7,8,10,12)(9,13,14,11)(15,16,18,22,28,25,30,27,20,17,19,23,21,24,29,26);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(15,16)(17,20)(18,26)(19,27)(21,25)(22,29)(23,30)(24,28);(4,6)(15,17,20,16)(18,27,19,26)(21,24,28,25)(22,29,23,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406721);