Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1790 |
| commutator | C2×Q8 | C22×C4×C5 |
| Frattini, cosocle | C22×Q8 | C23×C5 |
| prime radical | C5×#3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×Q8 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,12,18,13,19,11,17)(8,20,22,36,23,34,28,30)(9,24,26,15,10,16,25,33)(14,29,31,21,32,37,35,27);(6,8)(7,14)(9,21)(10,27)(11,28)(12,22)(13,23)(15,30)(16,34)(17,31)(18,35)(19,32)(20,24)(25,29)(26,37)(33,36);(6,9,13,10)(7,15,19,33)(8,21,23,27)(11,26,12,25)(14,30,32,36)(16,18,24,17)(20,31,34,35)(22,29,28,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5770);