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×#24930 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
862 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)(8,10)(9,11)(12,13)(14,15)(16,25)(17,30)(18,20)(19,24)(21,29)(22,31)(23,34)(26,35)(27,42)(28,37)(32,39)(33,38)(36,44)(40,41)(43,45);(6,8,9,12)(7,10,11,13)(15,20)(16,26)(18,34)(22,30)(23,24)(27,41)(28,29)(31,33)(35,37)(38,39)(42,45)(43,44);(7,11)(10,13)(14,16)(15,21)(17,27)(18,28)(19,29)(20,35)(22,36)(23,37)(24,25)(26,34)(30,45)(31,41)(32,43)(33,44)(38,40)(39,42);(14,17,19,32)(15,22,24,38)(16,27,29,43)(18,33,34,31)(20,30,23,39)(21,36,25,40)(26,41,28,44)(35,45,37,42)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17918);