Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C9×#473 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | #121 |
| commutator | C2×C4 | C23×C3×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C2×C32 | C3×#121 |
| prime radical | C32×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C2×C3×C4×C9 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,3)(13,14,19,24)(15,25,29,35)(16,30,32,38)(17,23,18,22)(20,28,36,27)(21,33,37,31)(26,40,42,34)(39,43,44,41);(4,5,6,7,8,9,10,11,12)(13,15,18,28,19,29,17,27)(14,20,23,35,24,36,22,25)(16,26,33,43,32,42,31,41)(21,34,38,44,37,40,30,39);(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,34)(22,30)(23,38)(24,37)(25,39)(27,41)(28,43)(29,42)(35,44)(36,40)432 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11240);