Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#12759 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
5740 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 5740 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4)(5,6)(7,8)(9,12,14,11)(10,13);(8,11)(9,14)(12,13);(4,6)(9,14)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231228, C2×#511771, #72231692, … |
| maximal normal subgroup of | #72231228, C2×#511771, #72231692, … |
| maximal quotient of | #72231142, #72231276, #72231282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400770);