Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C16 · Schur multiplier (H2) = C23×C8 |
| direct product | C8×C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C82 | C22 |
| commutator | C2 | C2×C8×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C4×C8 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
560 conjugacy classes of subgroups; only the 302 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 302 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,9,8,10,11,12);(1,2)(3,4)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(2,4)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35435);