Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C16 · Schur multiplier (H2) = C23×C42 |
| direct product | C16×#95 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C16 | C22 |
| commutator | C2 | C42×C16 |
| Frattini, cosocle | C23×C8 | C23 |
| socle, prime radical | C24 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
798 conjugacy classes of subgroups; only the 450 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 450 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(1,2)(3,4)(5,6,7,8);(2,4)(9,10,11,12)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53511);