Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3153 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
936 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)(7,10)(8,12)(11,13)(14,15)(16,26)(17,30)(18,25)(19,24)(20,23)(21,28)(22,31)(27,35)(29,38)(32,39)(33,36)(34,37)(40,42)(41,44)(43,45);(8,12)(10,13)(14,16)(15,21)(17,27)(18,33)(19,28)(20,29)(22,35)(23,36)(24,38)(25,26)(30,44)(31,42)(32,43)(34,41)(37,45)(39,40);(6,7)(8,13)(9,11)(10,12)(14,17,20,32)(15,22,23,37)(16,27,29,43)(18,31,19,34)(21,35,36,45)(24,30,25,39)(26,40,38,44)(28,41,33,42);(15,23)(21,36)(22,37)(24,25)(26,38)(30,39)(35,45)(40,44)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11982);