Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×C4×#389 |
| 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 | |
6114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6);(7,8)(9,10)(12,14)(15,16,18,17);(8,10)(15,17,18,16);(11,12,13,14)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#2428, C2×#676661, C2×#534553, … |
| maximal normal subgroup of | C2×C4×#2428, C2×#676661, C2×#534553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420549);