Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35248 |
| 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);(3,4,5,6)(8,11)(12,14)(19,27)(23,30)(25,26)(28,29);(3,4)(5,6)(15,16,20,24)(17,22,18,21)(19,23,27,30)(25,29,26,28);(7,8,10,11)(9,12,13,14)(15,17,20,18)(16,21,24,22)(19,25,27,26)(23,28,30,29);(15,18,20,17)(16,22,24,21)(19,26,27,25)(23,29,30,28);(7,9,10,13)(8,12,11,14)(15,19)(16,23)(17,25)(18,26)(20,27)(21,28)(22,29)(24,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535702);