Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25183 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11,16,22)(12,23,27,38)(13,29,14,19)(15,33,17,21)(18,28,35,26)(20,31,37,32)(24,36,40,39)(25,34,30,41);(6,8,9,7)(10,12,16,27)(11,18,22,35)(13,30,14,25)(15,28,17,26)(19,36,29,39)(20,34,37,41)(21,38,33,23)(24,32,40,31);(7,8)(10,13,16,14)(11,19,22,29)(12,24,27,40)(15,31,17,32)(18,34,35,41)(20,21,37,33)(23,39,38,36)(25,26,30,28);(10,14,16,13)(11,20,22,37)(12,25,27,30)(15,31,17,32)(18,34,35,41)(19,21,29,33)(23,36,38,39)(24,26,40,28)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18171);