Finite Group DatabasebetaFinite Group Databasebeta| order | 1539 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C19 · Schur multiplier (H2) = C3 |
| direct product | C19×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C19 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C32×C19 |
| prime radical | C19×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C19 ◁ C32×C19 ◁ G | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 513 |
A minimal faithful permutation representation — 2 generators on 46 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24)(22,28,32)(23,27,33)(25,34,30)(26,37,31)(29,39,42)(35,45,46)(36,44,41)(38,43,40);(20,22,29,23,30,40,31,41,46)(21,25,35,27,36,39,26,28,38)(24,32,42,33,34,43,37,44,45)323 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1539, 8);