Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28560 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(7,8)(9,13)(10,12)(11,14)(15,16)(17,25)(18,21)(19,24)(20,23)(22,28)(26,29)(27,30);(7,9,10,14)(8,11,12,13)(15,17,19,27,20,18,28,26)(16,21,24,29,23,25,22,30);(3,4,5,6)(15,18)(16,21)(17,20)(19,27)(22,29)(23,25)(24,30)(26,28);(8,12)(11,13)(15,19,20,28)(16,22,23,24)(17,26,18,27)(21,29,25,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366678);