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×#33770 |
| 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 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
576 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)(8,12)(9,11)(10,13)(14,15,18,24)(16,27,29,21)(17,26,20,23)(19,34,33,37)(22,40,41,30)(25,32,43,36)(28,38,31,42)(35,44,45,39);(6,8)(7,10)(9,13)(11,12)(14,16,18,29)(15,21,24,27)(17,31,20,28)(19,30,33,40)(22,34,41,37)(23,42,26,38)(25,39,43,44)(32,45,36,35);(8,13)(10,12)(14,16)(15,22)(17,28)(18,29)(19,35)(20,31)(21,37)(23,39)(24,41)(25,42)(26,44)(27,34)(30,36)(32,40)(33,45)(38,43);(7,11)(10,12)(14,17,18,20)(15,23,24,26)(16,28,29,31)(19,32,33,36)(21,38,27,42)(22,39,41,44)(25,37,43,34)(30,35,40,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26758);