Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19054 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3150 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
926 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)(7,10)(8,12)(11,13)(14,15)(16,26)(17,31)(18,23)(19,24)(20,25)(21,29)(22,33)(27,35)(28,36)(30,39)(32,37)(34,38)(40,42)(41,44)(43,45);(8,12)(10,13)(14,16)(15,21)(17,27)(18,28)(19,29)(20,30)(22,35)(23,26)(24,39)(25,36)(31,44)(32,41)(33,42)(34,43)(37,45)(38,40);(6,7)(8,13)(9,11)(10,12)(14,17,20,34)(15,22,25,37)(16,27,30,43)(18,33,19,32)(21,35,36,45)(23,38,24,31)(26,40,39,44)(28,42,29,41);(14,18,20,19)(15,23,25,24)(16,28,30,29)(17,32,34,33)(21,26,36,39)(22,31,37,38)(27,41,43,42)(35,44,45,40)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12042);