Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C22×C42 |
| direct product | C8×#338 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C42 |
| commutator | C4 | C42×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4×#95 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
696 conjugacy classes of subgroups; only the 346 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 346 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8);(9,10,12,16,14,17,13,22)(11,18,20,15,21,23,19,24);(9,11,13,20,14,21,12,19)(10,15,16,18,17,24,22,23)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12459);