Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C3 |
| direct product | C32×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C8 | D4 |
| commutator | C4 | C2×C32×C8 |
| Frattini, cosocle | C2×C8 | C22×C32 |
| socle | C2×C32 | C4⋊C8 |
| prime radical | C22×C32 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C8 ◁ C2×C32×C8 ◁ G | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8,11,16,12,15,20,22,13,17,19,14,10,21,9,18);(4,5,6)(7,9,10,19,13,20,12,11)(8,14,15,18,17,16,21,22)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1359);