Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×C4×#106 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle, prime radical | C25 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,7,8);(9,10,11,12)(14,17)(15,19)(18,20);(13,14,15,17,16,18,19,20)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#358973, C22×C4×#442, C2×#358697, … |
| maximal normal subgroup of | C2×#358973, C22×C4×#442, C2×#358697, … |
| maximal quotient of | C22×C4×#365, C2×#493137, C2×#183596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277105);