Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13269 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #391 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1985 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
308 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)(6,7,12,8,11,16,29,20,13,17,30,21,10,15,28,19)(9,22,25,34,24,14,31,18,26,32,36,23,27,33,37,35);(6,8,12,15,11,20,29,7,13,21,30,16,10,19,28,17)(9,23,36,33,24,35,37,22,26,34,25,14,27,18,31,32);(6,9,11,24,13,26,10,27)(7,14,15,22,17,33,16,32)(8,18,19,34,21,35,20,23)(12,25,29,31,30,36,28,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6257);