Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C2×C43 |
| direct product | C4×#1276 |
| subgroup | G/H | |
|---|---|---|
| center | C43 | D4 |
| commutator | C4 | C42×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C43 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1418 conjugacy classes of subgroups; only the 474 normal subgroups are shown.
This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 474 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(5,6)(7,9)(8,11)(10,12)(13,14,15,17,16,18,19,20);(6,9,11,10)224 conjugacy classes — view the full character table.
| maximal subgroup of | #72237307, #72237310, #72237311, … |
| maximal normal subgroup of | #72237307, #72237310, #72237311, … |
| maximal quotient of | #72237306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12461);