Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 82944 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C22×C3×C9×C16 |
| multiplicative | (ℤ/2261)ˣ, (ℤ/2907)ˣ, (ℤ/3213)ˣ, (ℤ/4522)ˣ, (ℤ/5814)ˣ, (ℤ/6426)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C8 | C23×C32 |
| socle, prime radical | C23×C32 | C3×C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(21,22,23,24,25,26,27,28,29);(17,18)(30,31,32);(19,20)1728 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11327);