Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26922 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7,9,10)(8,12,13,11)(14,15,18,23,19,24,34,21)(16,25,28,36,29,39,26,20)(17,30,31,38,33,37,32,22)(27,35,41,44,43,45,42,40);(7,10)(11,12)(14,16,19,29)(15,20,24,36)(17,27,33,43)(18,26,34,28)(21,25,23,39)(22,35,38,45)(30,40,37,44)(31,42,32,41);(15,21)(16,26)(18,34)(22,30)(23,24)(25,39)(27,41)(28,29)(31,32)(37,38)(40,44)(42,43);(6,8,9,13)(7,11,10,12)(14,17,19,33)(15,22,24,38)(16,27,29,43)(18,32,34,31)(20,35,36,45)(21,30,23,37)(25,40,39,44)(26,41,28,42)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19910);