Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3×C9×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8×C9 | C22 |
| commutator | C2 | C22×C3×C8×C9 |
| Frattini, cosocle | C3×C8 | C23×C32 |
| socle | C22×C32 | C2×C3×C8 |
| prime radical | C23×C32 | C3×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8×C9 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2)(19,20,21);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(22,23,24,25,26,27,28,29,30);(4,12)(6,14)(8,16)(10,18)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11330);