Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#23178 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,6)(7,8,10,9)(11,12,16,18,15,19,14,23)(13,17,21,26,22,20,24,25);(8,9)(11,12,14,23,15,19,16,18)(13,20,24,26,22,17,21,25);(3,4,5,6)(11,13)(12,17)(14,24)(15,22)(16,21)(18,26)(19,20)(23,25);(7,9,10,8)(11,14,15,16)(12,18,19,23)(13,21,22,24)(17,25,20,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289926);