Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12998 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1874 |
| prime radical | C5×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
500 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)(6,7,10,16,11,17,24,31,12,14,26,32,25,33,9,15)(8,18,21,28,22,34,19,35,20,36,23,13,27,29,37,30);(6,8,11,27,12,20,25,22)(7,13,17,35,14,28,33,30)(9,23,10,19,24,21,26,37)(15,34,16,18,31,29,32,36);(7,14)(9,24)(10,26)(13,28)(17,33)(19,37)(21,23)(30,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5986);