Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C9 · Schur multiplier (H2) = C26×C3 |
| direct product | C3×C9×#379 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C9 | C23 |
| commutator | C22 | C22×C3×C4×C9 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle | C23×C32 | C23×C3 |
| prime radical | C25×C32 | C2×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C9 ◁ G | |
2530 conjugacy classes of subgroups; only the 650 normal subgroups are shown.
This group has 2530 conjugacy classes of subgroups — too many to draw. Only its 650 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3)(14,16)(17,18,19,20)(22,24);(4,5,6,7,8,9,10,11,12)(21,22)(23,24);(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)756 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10958);