Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C9 · Schur multiplier (H2) = C2×C33 |
| direct product | C32×D4×C9 |
| unambiguous semidirect product | C4⋊(C2×C32×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C9 | C22 |
| commutator | C2 | C22×C32×C9 |
| Frattini, cosocle | C2×C3 | C22×C33 |
| socle | C2×C33 | C22×C3 |
| prime radical | C33×D4 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C9 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(16,17,18,19);(4,5,6)(17,19);(7,8,9,10,11,12,13,14,15)405 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 518);