Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12646 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #102 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | C4×#102 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
428 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)(6,7,10,17,13,19,12,29)(8,20,23,32,26,35,25,30)(9,15,27,34,28,18,11,16)(14,24,31,21,33,37,36,22);(6,8,11,24,12,25,28,22,13,26,27,37,10,23,9,21)(7,14,18,32,17,31,16,35,19,33,15,30,29,36,34,20);(6,9,10,27,13,28,12,11)(7,15,17,34,19,18,29,16)(8,21,23,37,26,22,25,24)(14,30,31,20,33,32,36,35)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5634);