Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#30570 |
| 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 | |
2628 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);(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16,20,19)(17,21,24,27)(18,22,26,28)(23,29,30,25);(7,9)(8,12)(10,13)(11,14)(15,17,20,24)(16,21,19,27)(18,25,26,29)(22,23,28,30);(4,6)(7,10,11,12)(8,9,13,14)(15,18)(16,22)(17,23)(19,28)(20,26)(21,29)(24,30)(25,27);(4,6)(8,13)(9,14)(15,16,20,19)(17,21,24,27)(18,22,26,28)(23,29,30,25)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408356);