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) = C26 |
| direct product | C5×#10999 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1310 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)(10,11,15,19,16,20,14,27)(12,21,23,17,24,32,28,31)(13,18,25,34,26,35,29,33)(22,37,40,41,39,36,38,30);(6,7)(8,9)(10,12)(11,17)(13,22)(14,28)(15,23)(16,24)(18,30)(19,32)(20,31)(21,27)(25,40)(26,39)(29,38)(33,36)(34,37)(35,41);(7,9)(10,13)(11,18)(12,22)(14,29)(15,25)(16,26)(17,30)(19,34)(20,35)(21,36)(23,40)(24,39)(27,33)(28,38)(31,41)(32,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3987);