Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×#368 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | D8 |
| commutator | C8 | C2×C32×C4 |
| Frattini, cosocle | C2×C8 | C22×C32 |
| socle | C2×C32 | #107 |
| prime radical | C22×C32 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C2×C32×C4 ◁ C2×C32×C8 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,2,3)(7,8,11,16,13,17,12,27)(9,18,21,33,23,34,22,37)(10,24,25,31,26,38,19,15)(14,28,29,36,30,35,32,20);(4,5,6)(7,9,19,35,12,22,26,36,13,23,25,28,11,21,10,20)(8,14,24,33,16,29,31,34,17,30,38,37,27,32,15,18)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1363);