Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33829 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#2218 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(7,10)(8,12)(14,15,17,22)(16,25,26,20)(18,21,29,37)(19,32,30,24)(23,39,38,31)(27,33,42,36)(28,35,43,41)(34,45,40,44);(6,7)(8,11)(9,10)(12,13)(14,16,17,26)(15,20,22,25)(18,27,29,42)(19,28,30,43)(21,36,37,33)(23,34,38,40)(24,35,32,41)(31,45,39,44);(7,10)(11,13)(15,21)(16,26)(18,31)(19,30)(20,33)(22,37)(23,24)(25,36)(27,44)(29,39)(32,38)(34,41)(35,40)(42,45);(6,8,9,12)(7,11,10,13)(15,22)(16,26)(21,37)(23,38)(24,32)(27,42)(28,43)(44,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26817);