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×#35244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8204 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)(7,8,10,12)(9,11,13,14)(15,16,21,26)(17,25,20,22)(18,19,24,23)(27,30,29,28);(7,8)(9,11)(10,12)(13,14)(15,17,21,20)(16,22,26,25)(18,29,24,27)(19,30,23,28);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,18,21,24)(16,23,26,19)(17,27,20,29)(22,30,25,28);(15,19,21,23)(16,24,26,18)(17,28,20,30)(22,29,25,27);(7,9,10,13)(8,11,12,14)(15,20,21,17)(16,25,26,22)(18,29,24,27)(19,30,23,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535696);