Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C22×#1808 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6354 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)(14,17)(15,18)(19,20);(13,14,18,20,16,19,15,17);(6,8)176 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#21528, C22×C22⋊C4×D8, C22×#22659, … |
| maximal normal subgroup of | C22×#21528, C22×C22⋊C4×D8, C22×#22659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278001);