Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#13999 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7998 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);(4,6)(8,10)(15,16,18,17);(3,4)(5,6)(11,12,13,14)(15,17,18,16);(7,8,9,10)(11,12)(13,14);(3,4)(5,6)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#873825, C2×#773894, C2×#532452, … |
| maximal normal subgroup of | C2×#873825, C2×#773894, C2×#532452, … |
| maximal quotient of | C2×#154727, C2×#154522, C2×#155122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421487);