Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3×C9 · Schur multiplier (H2) = C29×C3 |
| direct product | C23×C3×Q8×C9 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C3×C9 | C22 |
| commutator | C2 | C25×C3×C9 |
| Frattini, cosocle | C2×C3 | C25×C32 |
| socle, prime radical | C24×C32 | C22×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3×C9 ◁ G | |
4250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2)(15,16,17);(3,4)(18,19,20,21,22,23,24,25,26);(5,6);(7,8,9,10)(11,14,13,12);(7,11,9,13)(8,12,10,14)1080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31080);