Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#35308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#537 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(4,6)(8,9)(11,12,16,18)(13,19,20,14)(15,23,24,17)(21,22,26,25);(7,8,10,9)(13,20)(14,19)(21,26)(22,25);(3,4)(5,6)(7,9,10,8)(11,13)(12,14)(15,21)(16,20)(17,22)(18,19)(23,25)(24,26);(11,14)(12,13)(15,22)(16,19)(17,21)(18,20)(23,26)(24,25);(11,15,16,24)(12,17,18,23)(13,21,20,26)(14,22,19,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535785);