Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C9×#509 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C2×D8 |
| commutator | C8 | C23×C3×C9 |
| Frattini, cosocle | C3×C8 | C23×C32 |
| socle | C2×C32 | C2×C3×D8 |
| prime radical | C32×#509 | C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C2×C3×C4×C9 ◁ C2×C3×C8×C9 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3)(13,14)(15,18)(16,25)(17,19)(20,22)(21,23)(24,26)(27,28);(4,5,6,7,8,9,10,11,12)(14,18)(15,22)(16,23)(19,26)(20,21)(24,28)(25,27);(14,19)(18,26)(20,27)(21,25)432 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11348);