Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C9×SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | D16 |
| commutator | C16 | C22×C3×C9 |
| Frattini, cosocle | C3×C16 | C22×C32 |
| socle | C2×C32 | C3×D16 |
| prime radical | C32×D16 | C2×C3 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C3×C4×C9 ◁ C3×C8×C9 ◁ C3×C9×C16 ◁ G | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44);(4,5,6,7,8,9,10,11,12)(14,28)(15,43)(16,26)(17,41)(18,24)(19,39)(20,22)(21,37)(23,35)(25,33)(27,31)(30,44)(32,42)(34,40)(36,38)513 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2180);