Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#26141 |
| 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 | |
2624 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)(9,12)(10,14)(11,13)(15,16,19,20)(17,21,24,28)(18,25,27,22)(23,26,30,29);(4,6)(7,9,11,10)(8,12,13,14)(15,17)(16,21)(18,26)(19,24)(20,28)(22,23)(25,30)(27,29);(7,10,11,9)(8,12,13,14)(15,18)(16,22)(17,23)(19,27)(20,25)(21,29)(24,30)(26,28);(4,6)(15,16,19,20)(17,21,24,28)(18,22,27,25)(23,29,30,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345704);