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×#578 |
| 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 | |
3472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3)(10,11,15,19)(12,20,23,33)(13,17,24,31)(14,26,28,18)(16,27,30,22)(21,34,35,41)(25,38,37,32)(29,39,40,36);(4,5,6)(10,12,15,23)(11,16,19,30)(13,21,24,35)(14,27,28,22)(17,29,31,40)(18,33,26,20)(25,39,37,36)(32,41,38,34);(7,8,9)(10,13)(11,17)(12,21)(14,25)(15,24)(16,29)(18,32)(19,31)(20,34)(22,36)(23,35)(26,38)(27,39)(28,37)(30,40)(33,41);(13,24)(17,31)(21,35)(25,37)(29,40)(32,38)(34,41)(36,39)594 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46096);