Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C9×#332 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C22⋊C4 |
| commutator | C2×C4 | C2×C3×C4×C9 |
| Frattini, cosocle | C3×C42 | C22×C32 |
| socle, prime radical | C22×C32 | C3×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C3×C42×C9 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3)(14,17)(15,19)(18,20)(21,22,24,26,25,28,23,27);(4,5,6,7,8,9,10,11,12)(13,14,16,18)(15,20,19,17)(22,26)(23,24)(27,28)513 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2060);