Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33×C4 · Schur multiplier (H2) = C25×C33 |
| direct product | C33×#568 |
| subgroup | G/H | |
|---|---|---|
| center | C33×C4 | C24 |
| commutator | C2 | C23×C33×C4 |
| Frattini, cosocle | C4 | C24×C33 |
| socle | C2×C33 | C23×C4 |
| prime radical | C2×C33×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C33×C4 ◁ G | |
3612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3)(10,11,13,17,14,16,20,23)(12,15,19,22,18,21,24,25);(4,5,6)(10,12)(11,15)(13,19)(14,18)(16,21)(17,22)(20,24)(23,25);(7,8,9)(12,18)(15,21)(19,24)(22,25);(11,16)(15,21)(17,23)(22,25)918 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46086);