Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C27 · Schur multiplier (H2) = C22 |
| direct product | C27×#331 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C27 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C27 |
| Frattini, cosocle | C42×C9 | C22×C3 |
| socle | C22×C3 | C9×C22⋊C4 |
| prime radical | C3×#127 | C2×C9 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C27 ◁ C42×C27 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,33,31,32,34,35)(36,37)(38,40)(39,42)(41,43);(29,32)(30,34)(37,40)(38,43)(41,42)513 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 137);