Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#36699 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,16)(13,17)(14,15)(18,21)(19,23)(20,22)(24,25)(26,27,30,33)(28,31,35,39)(29,32,37,41)(34,38,36,40);(11,15)(16,21)(17,22)(23,25)(26,28,30,35)(27,31,33,39)(29,36,37,34)(32,40,41,38);(7,9)(10,12)(11,16)(13,19)(14,18)(15,21)(17,23)(20,24)(22,25)(26,29)(27,32)(28,34)(30,37)(31,38)(33,41)(35,36)(39,40);(12,18)(16,21)(19,24)(23,25)(27,33)(29,37)(31,39)(34,36);(10,13,14,20)(11,17,15,22)(12,19,18,24)(16,23,21,25)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29687);