Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33876 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
564 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)(16,25)(17,22)(18,30)(19,34)(20,24)(21,28)(23,32)(26,41)(27,43)(29,40)(31,37)(33,42)(35,39)(36,44)(38,45);(6,8)(7,10)(9,13)(11,12)(16,26)(21,36)(25,41)(27,35)(28,44)(31,45)(37,38)(39,43);(14,16,20,26)(15,21,24,36)(17,27,19,35)(18,31,33,45)(22,37,34,38)(23,39,40,43)(25,42,41,30)(28,29,44,32);(8,13)(10,12)(14,17,20,19)(15,22,24,34)(16,27,26,35)(18,29,33,32)(21,37,36,38)(23,30,40,42)(25,43,41,39)(28,31,44,45)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26864);