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×#26946 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2668 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)(9,13)(10,12)(11,14)(15,16)(17,25)(18,22)(19,24)(20,23)(21,27)(26,29)(28,30);(3,4,6,5)(7,9,10,14)(8,11,12,13)(15,17)(16,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,25);(8,12)(11,13)(15,18)(16,22)(19,20)(23,24)(26,28)(29,30);(3,5,6,4)(8,12)(11,13)(15,18,19,20)(16,23,24,22)(17,26,27,28)(21,29,25,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350734);