Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#33435 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,20,15,21,29,37,16,19,30,35,31,36,32,41)(12,17,23,33,24,13,25,26,22,27,38,40,39,28,18,34);(7,9)(10,12)(11,17)(13,21)(14,23)(15,24)(16,22)(18,32)(19,27)(20,33)(25,29)(26,37)(28,36)(30,38)(31,39)(34,41)(35,40);(10,13,14,26,15,27,29,40,16,28,30,34,31,17,32,33)(11,18,20,12,21,23,37,24,19,25,35,22,36,38,41,39);(11,19)(12,22)(18,25)(20,35)(21,36)(23,38)(24,39)(37,41)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26423);