Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×C8×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6,7,8,9,10);(12,14)(16,17);(15,16,18,17);(11,12,13,14)(15,17,18,16)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C8×D42, C2×#210434, C2×#212800, … |
| maximal normal subgroup of | C2×C8×D42, C2×#210434, C2×#212800, … |
| maximal quotient of | C2×#212800 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280900);