Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C32×#427 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | D4 |
| commutator | C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C4 | C23×C32 |
| socle | C22×C32 | C2×D4 |
| prime radical | C23×C32 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C22×C32×C4 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(7,8,9,10)(11,12)(13,20)(14,18)(15,24)(16,19)(17,22)(21,25)(23,26);(4,5,6)(11,13,15,22,16,23,14,21)(12,17,18,20,19,25,24,26);(11,14,16,15)(12,18,19,24)(13,21,23,22)(17,20,25,26)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 4882);