Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#35320 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(9,13)(11,14)(15,16,20,23)(17,18,21,26)(19,27,28,22)(24,30,29,25);(3,4,5,6)(7,8,10,12)(9,11,13,14)(15,17)(16,18)(19,24)(20,21)(22,25)(23,26)(27,30)(28,29);(7,9,10,13)(8,11,12,14)(15,18)(16,21)(17,23)(19,25)(20,26)(22,29)(24,27)(28,30);(15,18)(16,21)(17,23)(19,25)(20,26)(22,29)(24,27)(28,30);(15,19,20,28)(16,22,23,27)(17,24,21,29)(18,25,26,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535800);