Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 4644864 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C26×C3 |
| abelian | C23×C3×C8×C9 |
| multiplicative | (ℤ/4088)ˣ, (ℤ/4256)ˣ, (ℤ/5256)ˣ, (ℤ/5472)ˣ, (ℤ/6048)ˣ, (ℤ/6132)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C4 | C24×C32 |
| socle, prime radical | C24×C32 | C3×C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1728 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3,4,5,6,7,8)(15,16,17,18,19,20,21,22,23);(9,10)(24,25,26);(11,12);(13,14)1728 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20740);