Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
5444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12,14,16,15,18,13,17);(4,7)(5,9)(12,16)(13,14)(17,18);(4,7)(5,9)86 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#185755, C2×#245712, C2×#245711, … |
| maximal normal subgroup of | C2×#185755, C2×#245712, C2×#245711, … |
| maximal quotient of | #72257804, #72257851, #72261311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92899);