Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C8 · Schur multiplier (H2) = C22×C44 |
| direct product | C8×#376 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C8 | C22 |
| commutator | C2 | C2×C42×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups; only the 1234 normal subgroups are shown.
This group has 1882 conjugacy classes of subgroups — too many to draw. Only its 1234 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:
(17,18,19,21,20,22,23,24);(1,2,3,4)(9,10)(11,14)(12,13)(15,16);(5,6,7,8)(10,13)(14,16);(9,11,12,15)(10,14,13,16)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260362);