Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C4×#2329 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3578 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);(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23);(5,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,20)(18,22,23,24);(11,16)(13,15)(17,20,21,19)(18,22,23,24);(10,14)(13,15)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420625);