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×#436 |
| 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 | C2×C3×D4 | C3×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
183 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 183 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) = 21 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,18,16,17,19,21,20);(11,13)(15,19)(16,20);(14,16,18,19,17,20,21,15)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 872);