Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#10956 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(10,11)(12,21)(13,25,27,34)(14,28)(15,19)(16,20)(17,23)(18,26,35,30)(22,37,40,31)(24,32)(29,33)(36,39,41,38);(6,7,8,9)(10,12)(11,17)(13,22)(14,29)(15,23)(16,24)(18,31)(19,32)(20,33)(21,28)(25,36)(26,39)(27,40)(30,38)(34,41)(35,37);(10,13,15,30,16,27,14,26)(11,18,19,34,20,35,28,25)(12,22,23,38,24,40,29,39)(17,31,32,41,33,37,21,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3944);