Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | D4 |
| commutator | C4 | C42×C9 |
| Frattini, cosocle | C22×C3×C4 | C22×C3 |
| socle | C22×C3 | C3×C4⋊C4 |
| prime radical | C23×C3 | C3×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C22×C4×C9 ◁ G | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,17,22,19,23,18,21)(16,24,25,20,27,28,26,29);(11,13)(14,16,18,26,19,27,17,25)(15,20,22,28,23,29,21,24)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 147);