Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#17008 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,5,6)(8,11)(9,13)(16,19)(17,21)(24,26)(28,29);(8,11)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24,27,26)(25,29,30,28);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(15,16)(17,22)(18,19)(20,21)(23,25,27,30)(24,28,26,29);(7,9)(8,12)(10,13)(11,14)(15,17,18,21)(16,20,19,22)(23,26,27,24)(25,28,30,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 519116);