Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#12002 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #106 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C2×C5 | C2×#334 |
| prime radical | C23×C5 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
220 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)(6,7,10,16,12,17,27,33,13,18,26,15,11,29,9,25)(8,19,22,31,24,34,21,30,20,32,28,37,23,36,35,14);(6,8,9,21,11,23,26,22,13,20,27,35,12,24,10,28)(7,14,15,19,17,31,25,34,18,30,16,32,29,37,33,36);(8,20)(14,30)(19,32)(21,35)(22,28)(23,24)(31,37)(34,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4990);