Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#2584 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2);(3,4);(5,6,7,8)(9,10,12,14)(11,15,16,13)(18,20)(22,23);(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,23,24,22);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23);(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,23,24,22)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535255);