Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C9 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C3×C9×#427 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C9 | D4 |
| commutator | C4 | C22×C3×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C22×C32 | C2×C3×D4 |
| prime radical | C23×C32 | C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C9 ◁ C22×C3×C4×C9 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3)(13,14,15,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(4,5,6,7,8,9,10,11,12)(17,19,21,28,22,29,20,27)(18,23,24,26,25,31,30,32);(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11102);