Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C9×#508 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C9 | D8 |
| commutator | C8 | C23×C3×C9 |
| Frattini, cosocle | C3×C8 | C23×C32 |
| socle | C2×C32 | C2×C3×D8 |
| prime radical | C32×#508 | C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C9 ◁ C2×C3×C4×C9 ◁ C2×C3×C8×C9 ◁ G | |
420 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)(15,24)(16,21)(17,29)(18,33)(19,23)(20,26)(22,31)(25,40)(27,41)(28,39)(30,35)(32,43)(34,38)(36,44)(37,42);(4,5,6,7,8,9,10,11,12)(13,15)(14,20)(16,25)(17,30)(18,34)(19,27)(21,35)(22,38)(23,37)(24,29)(26,31)(28,42)(32,44)(33,36)(39,40)(41,43);(13,16,19,18)(14,21,23,33)(15,25,27,34)(17,28,32,31)(20,35,37,36)(22,29,39,43)(24,40,41,38)(26,30,42,44)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11345);