Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26790 |
| 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 | |
2370 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);(4,5)(7,8)(9,14)(10,12)(11,13)(15,16)(17,23)(18,21)(19,22)(20,25)(24,29)(26,28)(27,30);(3,4,6,5)(7,9,11,10)(8,12,13,14)(15,17)(16,20)(18,25)(19,24)(21,23)(22,28)(26,27)(29,30);(3,4,6,5)(7,10,11,9)(8,12,13,14)(17,24)(19,27)(20,28)(22,30)(23,29)(25,26);(3,5,6,4)(8,13)(12,14)(16,21)(20,23)(22,30)(28,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350953);