Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C82 · Schur multiplier (H2) = C27×C4 |
| direct product | C22×#1273 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C42 | C22 |
| commutator | C2 | C22×C82 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C25 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2954 conjugacy classes of subgroups; only the 1922 normal subgroups are shown.
This group has 2954 conjugacy classes of subgroups — too many to draw. Only its 1922 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2);(3,4);(5,6)(7,8)(9,10,11,13,12,14,15,16);(6,8)(17,18,19,21,20,22,23,24)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×#1273⋊C3, C2×#105250, C22×#10304, … |
| maximal normal subgroup of | C22×#1273⋊C3, C2×#105250, C22×#10304, … |
| maximal quotient of | C22×#7014, C22×#7137 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88042);