Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3061 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
532 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)(10,11,16,22)(12,23,26,39)(13,27,31,38)(14,21,15,20)(17,33,36,24)(18,32,37,25)(19,29,34,30)(28,40,41,35);(6,7)(8,9)(10,12)(11,17)(13,28)(14,25)(15,32)(16,26)(18,34)(19,37)(20,35)(21,40)(22,36)(23,27)(24,30)(29,33)(31,41)(38,39);(7,9)(10,12)(11,18)(13,24)(14,32)(15,25)(16,26)(17,34)(19,36)(20,39)(21,23)(22,37)(27,40)(28,30)(29,41)(31,33)(35,38)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6296);