Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C42×#123 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4);(5,6,7,8);(9,10)(11,12)(13,14)(15,18)(16,17)(19,20);(14,17)(18,20);(10,12)(13,15,16,19)(14,18,17,20)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420566);