Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19097 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3567 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(8,15)(9,10)(12,19)(13,17)(16,18)(20,21)(22,23,28,33)(24,34,36,46)(25,38,40,47)(26,32,27,31)(29,41,44,37)(30,42,45,39)(35,43,49,53)(48,50,52,51);(6,7)(8,12)(9,13)(10,17)(11,14)(15,20)(16,21)(18,19)(22,24,28,36)(23,29,33,44)(25,35,40,49)(26,41,27,37)(30,43,45,53)(31,46,32,34)(38,52,47,48)(39,51,42,50);(6,8)(7,12)(9,15)(10,18)(11,16)(13,20)(14,21)(17,19)(22,25)(23,30)(24,35)(26,39)(27,42)(28,40)(29,43)(31,38)(32,47)(33,45)(34,48)(36,49)(37,50)(41,51)(44,53)(46,52);(6,9,11,10)(7,13,14,17)(8,15,16,18)(12,20,21,19)(24,36)(29,44)(34,46)(35,49)(37,41)(43,53)(48,52)(50,51)155 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12085);