Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C26 |
| direct product | C22×C4⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C16 | C22 |
| commutator | C2 | C23×C32 |
| Frattini, cosocle | C2×C16 | C24 |
| socle, prime radical | C24 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
542 conjugacy classes of subgroups; only the 442 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 442 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2);(3,4);(6,8)(9,10,11,15,12,16,19,25,13,17,20,26,22,28,31,35,14,18,21,27,23,29,32,36,24,30,33,37,34,38,39,40);(5,6,7,8)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419900);