Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12842 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,19,16,13,27,20)(12,21,25,36,26,38,41,37)(14,29,22,18,30,34,39,28)(17,31,32,35,23,40,33,24);(10,12,22,31,15,25,30,35,16,26,39,40,27,41,14,24)(11,17,29,37,20,33,28,38,13,23,34,36,19,32,18,21);(7,9)(10,13)(11,16)(12,23)(14,28)(15,19)(17,26)(18,30)(20,27)(21,35)(22,34)(24,38)(25,32)(29,39)(31,36)(33,41)(37,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5830);