Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3171 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15)(9,13)(10,19)(11,14)(12,17)(16,20)(18,21)(24,30)(25,32)(27,35)(28,33)(31,34)(36,37);(6,8)(7,12)(9,17)(10,16)(11,18)(13,21)(14,20)(15,19)(22,23,26,29)(24,27,31,36)(25,33,32,28)(30,37,34,35);(6,7)(8,12)(9,13)(10,19)(11,14)(15,16)(17,21)(18,20)(22,24)(23,27)(25,34)(26,31)(28,37)(29,36)(30,32)(33,35);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,15,20,21)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12019);