Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2332 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)(7,8,11,13)(9,12,10,14)(15,16,20,24)(17,21,27,30)(18,22,19,23)(25,28,26,29);(4,6)(7,9,11,10)(8,12,13,14)(15,17,20,27)(16,21,24,30)(18,26,19,25)(22,29,23,28);(7,10,11,9)(8,12,13,14)(17,25)(18,19)(21,28)(22,23)(26,27)(29,30);(4,6)(15,18,20,19)(16,22,24,23)(17,26,27,25)(21,29,30,28)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346368);