Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×#1965 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#127 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
8202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(6,8)(9,10,12,15,13,14,16,11);(9,11,12,14,13,15,16,10);(5,6,7,8)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#369039, C2×#6443311, C2×#361625, … |
| maximal normal subgroup of | C2×#369039, C2×#6443311, C2×#361625, … |
| maximal quotient of | C22×#10981 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279169);