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×#19034 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
836 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)(7,12)(8,15)(9,10)(14,17)(16,18)(19,21)(22,23,27,25)(24,30,32,37)(26,34,36,29)(28,35,31,33);(6,7)(8,13)(9,17)(10,12)(11,14)(15,21)(16,20)(18,19)(22,24,27,32)(23,28,25,31)(26,35,36,33)(29,37,34,30);(6,8,11,16)(7,13,14,20)(9,15,10,18)(12,19,17,21)(22,25,27,23)(24,31,32,28)(26,29,36,34)(30,35,37,33);(7,14)(12,17)(13,20)(19,21)(24,32)(28,31)(30,37)(33,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12022);