Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20986 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5×Q8 ◁ G | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,16,22,29)(13,17)(14,18)(15,19)(20,27,35,39)(21,28,37,41)(23,30)(24,31)(25,32)(26,33)(34,38,36,40);(10,12)(11,16,19,29)(13,20)(14,21)(15,22)(17,27,31,39)(18,28,33,41)(23,34)(24,35)(25,36)(26,37)(30,38,32,40);(7,9)(10,13,15,24)(11,17,19,31)(12,20,22,35)(14,25,26,23)(16,27,29,39)(18,32,33,30)(21,36,37,34)(28,40,41,38);(10,14,15,26)(11,18,19,33)(12,21,22,37)(13,23,24,25)(16,28,29,41)(17,30,31,32)(20,34,35,36)(27,38,39,40)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13974);