Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C22×C3×C42 |
| direct product | C32×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C32 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8,11,14)(9,16,17,13)(10,18,20,22)(12,15,21,19);(4,5,6)(7,9,11,17)(8,13,14,16)(10,19,20,15)(12,18,21,22);(7,10)(8,9)(11,20)(12,13)(14,17)(15,18)(16,21)(19,22);(7,9)(8,10)(11,17)(12,18)(13,15)(14,20)(16,19)(21,22)171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8244);