Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13140 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #481 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1926 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
314 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)(6,7,12,18,10,16,13,28)(8,17,11,26,20,14,21,30)(9,22,23,33,25,37,24,29)(15,27,31,19,32,35,34,36);(6,8,13,21,10,20,12,11)(7,14,28,26,16,17,18,30)(9,19,23,35,25,36,24,27)(15,29,31,22,32,33,34,37);(6,9,12,24,10,25,13,23)(7,15,28,31,16,32,18,34)(8,19,11,27,20,36,21,35)(14,29,26,22,17,33,30,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6128);