Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C42 |
| direct product | C2×#21260 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
8330 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);(4,6)(8,10)(11,12,15,17,16,13,14,18);(3,4,5,6)(11,13,16,12)(14,18,15,17);(7,8,9,10);(11,14,16,15)(12,17,13,18)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#418, C2×#6443168, C2×#6442515, … |
| maximal normal subgroup of | C2×D4×#418, C2×#6443168, C2×#6442515, … |
| maximal quotient of | C2×#357498, #72302344, #72305868 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279240);