Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C42 |
| direct product | C2×C4×#387 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| socle, prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,5,6);(7,8)(9,10)(11,12,14,13)(15,16)(17,20)(18,19)(21,22);(11,13,14,12)(16,19)(20,22);(8,10)(15,17,18,21)(16,20,19,22)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420546);