Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×D4⋊C8 |
| unambiguous semidirect product | D4⋊(C8×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | D4 |
| commutator | C4 | C2×C8×C9 |
| Frattini, cosocle | C3×C42 | C22×C3 |
| socle | C22×C3 | C3×C22⋊C4 |
| prime radical | C2×C3×D4 | C3×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C42×C9 ◁ G | |
123 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 123 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,14)(13,16)(15,17)(18,19,20,22,21,23,24,25);(11,14)(12,17)(15,16)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 131);