Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11065 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,15,19,16,21,14,20)(12,17,24,32,25,33,23,37)(13,26,28,18,29,35,30,36)(22,34,38,31,39,41,27,40);(6,7,8,9)(10,12)(11,17)(13,27)(14,23)(15,24)(16,25)(18,34)(19,32)(20,37)(21,33)(22,28)(26,40)(29,38)(30,39)(31,35)(36,41);(10,13)(11,18)(12,22)(14,30)(15,28)(16,29)(17,31)(19,35)(20,26)(21,36)(23,27)(24,38)(25,39)(32,41)(33,40)(34,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4053);