Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
| direct product | C8×#533 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C23 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4,5,6,7,8);(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,23,24,21);(10,13)(14,16);(17,19,20,24)(18,21,22,23);(9,11,12,15)(10,14,13,16)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281251);