Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#20747 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,14,17,15,13,18,16);(4,7)(8,10);(11,13)(12,15)(14,16)(17,18);(3,5,6,9)(4,8,7,10)152 conjugacy classes — view the full character table.
| maximal subgroup of | #72239472, #72239502, #72239503, … |
| maximal normal subgroup of | #72239472, #72239502, #72239503, … |
| maximal quotient of | #72239496, C2×#191008, C2×#191007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278190);