Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#421 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | C2×D4 |
| commutator | C4 | C22×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C3 |
| socle | C2×C3 | C2×C3×C22⋊C4 |
| prime radical | C2×C3×#41 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C22×C4×C9 ◁ G | |
231 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11,14,17,15,18,13,22)(12,19,21,16,20,25,23,24);(10,12)(11,16)(13,23)(14,21)(15,20)(17,25)(18,24)(19,22);(12,20)(16,24)(19,25)(21,23)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 857);