Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#477 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C2×D4 |
| commutator | C2×C4 | C23×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C3 |
| socle, prime radical | C22×C3 | C2×C3×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C42×C9 ◁ G | |
144 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 576 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,15)(12,16,17,14)(19,22)(20,23)(24,25);(10,12,13,17)(11,14,15,16)(18,19,21,22)(20,23,24,25);(18,20,21,24)(19,23,22,25)171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 913);