Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33818 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
490 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)(14,15,19,24)(16,20,25,35)(17,21,30,38)(18,31,29,23)(22,39,37,28)(26,36,40,32)(27,44,41,34)(33,42,45,43);(6,7)(8,11)(9,10)(12,13)(14,16)(15,20)(17,26)(18,27)(19,25)(21,36)(22,33)(23,34)(24,35)(28,43)(29,41)(30,40)(31,44)(32,38)(37,45)(39,42);(7,10)(11,13)(15,21)(16,25)(17,28)(18,29)(20,32)(22,23)(24,38)(26,42)(30,39)(31,37)(33,44)(34,45)(35,36)(40,43);(6,8,9,12)(7,11,10,13)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26806);