Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3283 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
546 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)(6,7,9,11)(8,12,13,10)(14,15)(16,21)(17,27)(18,30)(19,34)(20,24)(22,28)(23,32)(25,41)(26,40)(29,43)(31,36)(33,44)(35,39)(37,45)(38,42);(6,8)(7,10)(9,13)(11,12)(14,16,20,19)(15,21,24,34)(17,25,29,35)(18,26,33,32)(22,36,38,37)(23,30,40,44)(27,41,43,39)(28,31,42,45);(14,17)(15,22)(16,25)(18,31)(19,35)(20,29)(21,36)(23,39)(24,38)(26,42)(27,30)(28,32)(33,45)(34,37)(40,41)(43,44);(8,13)(10,12)(14,16,20,19)(15,21,24,34)(17,25,29,35)(18,26,33,32)(22,36,38,37)(23,30,40,44)(27,41,43,39)(28,31,42,45)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26857);