Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#13223 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#2541 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(4,6)(7,8)(9,13,14,10)(11,12);(7,9,13,12)(8,11,14,10);(3,4,5,6)74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#502798, C2×#500507, C2×D4×#353, … |
| maximal normal subgroup of | C2×#502798, C2×#500507, C2×D4×#353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402365);