Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C33 · Schur multiplier (H2) = C25×C33 |
| direct product | C33×#577 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24×C33 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C33×#41 ◁ 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)(10,11,14,18)(12,16,19,23)(13,15,21,17)(20,22,25,24);(4,5,6)(11,15)(13,21)(16,22)(17,18)(20,25)(23,24);(7,8,9)(10,12)(11,16)(13,20)(14,19)(15,22)(17,24)(18,23)(21,25);(12,19)(16,23)(20,25)(22,24)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46095);