Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20962 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| 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×C4×C5 ◁ G | |
1742 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)(7,9)(10,11)(12,16)(13,23,25,17)(14,26)(15,19)(18,27)(20,28,33,36)(21,34)(22,30)(24,31,38,37)(29,35)(32,40,41,39);(6,7,8,9)(10,12,15,22)(11,16,19,30)(13,20,25,33)(14,21,27,35)(17,28,23,36)(18,29,26,34)(24,32,38,41)(31,39,37,40);(10,13)(11,17)(12,20)(14,24)(15,25)(16,28)(18,31)(19,23)(21,32)(22,33)(26,37)(27,38)(29,39)(30,36)(34,40)(35,41);(10,14,15,27)(11,18,19,26)(12,21,22,35)(13,24,25,38)(16,29,30,34)(17,31,23,37)(20,32,33,41)(28,39,36,40)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13950);