Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#10951 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1437 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12,17,27)(9,19)(10,22)(11,14)(13,21)(15,24)(16,30,32,33)(18,29,23,26)(20,34)(25,36,37,31)(28,35);(6,8)(7,12,14,27)(9,16)(10,23)(11,17)(13,25,28,37)(15,29,22,26)(18,24)(19,33,34,30)(20,32)(21,36)(31,35);(6,9,10,21,11,20,24,35)(7,13,15,19,14,28,22,34)(8,16,23,36,17,32,18,31)(12,25,29,33,27,37,26,30)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3939);