Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#432 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C23 |
| commutator | C22 | C22×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C3 |
| socle | C22×C3 | C22×C3×C4 |
| prime radical | C23×C3 | C2×C3×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
165 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,15)(13,14)(16,17)(18,19,20,23,21,22,24,25);(11,14)(15,17);(10,12,13,16)(11,15,14,17)(19,22)(23,25)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 868);