Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C33 · Schur multiplier (H2) = C29×C33 |
| direct product | C33×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C33×C4 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C33 |
| socle | C2×C33 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C33×C4 ◁ G | |
11340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2,3)(10,11)(12,18)(13,17)(14,16)(15,19)(20,24)(21,23)(22,25);(4,5,6)(10,12)(11,15)(13,20)(14,19)(16,18)(17,22)(21,25)(23,24);(7,8,9)(12,19)(15,18)(20,25)(22,24);(11,16)(15,18)(17,23)(22,24);(10,13,14,21)(11,17,16,23)(12,20,19,25)(15,22,18,24)918 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47839);