Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×#560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24 |
| direct product | 3−1+2×#564 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22×C3 | C24×C32 |
| prime radical | C22×C32 | C24×C3 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,4,3,6,8,7,5,9)(12,18)(13,21)(15,19)(16,23)(26,27,30,33)(28,31,36,34)(29,37,38,32)(35,40,41,39);(2,5,6)(4,8,9)(10,11,14,17)(12,19,18,15)(13,16,21,23)(20,25,24,22)(26,27)(28,34)(29,37)(30,33)(31,36)(32,38)(35,39)(40,41);(10,12,14,18)(11,15,17,19)(13,20,21,24)(16,22,23,25)(26,28)(27,31)(29,35)(30,36)(32,39)(33,34)(37,40)(38,41);(10,13)(11,16)(12,20)(14,21)(15,22)(17,23)(18,24)(19,25)(26,29,30,38)(27,32,33,37)(28,35,36,41)(31,39,34,40)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20739);