Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12703 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4106 conjugacy classes of subgroups; only the 338 normal subgroups are shown.
This group has 4106 conjugacy classes of subgroups — too many to draw. Only its 338 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,11,16,12,17,9,14)(10,13,18,21,19,15,20,22);(8,13)(9,11)(10,19)(14,22)(15,17)(16,21);(4,6)(7,9,12,11)(8,14,17,16)(10,18,19,20)(13,21,15,22)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#493385, C2×#94425, C2×#133350, … |
| maximal normal subgroup of | C2×#493385, C2×#94425, C2×#133350, … |
| maximal quotient of | #72261807, C2×#134109, #72293127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400576);