Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C32×#493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3)(8,10)(12,15)(13,16)(17,18);(4,5,6)(11,12,16,18,14,17,13,15);(7,8,9,10)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4948);