Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23017 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×C4⋊C8 ◁ G | |
864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11,15,20,16,19,14,28)(12,17,23,32,24,31,22,38)(13,25,26,18,27,34,29,33)(21,35,36,30,37,41,39,40);(7,8)(10,12,15,23,16,24,14,22)(11,17,20,32,19,31,28,38)(13,21,26,36,27,37,29,39)(18,30,34,41,33,40,25,35);(10,13)(11,18)(12,21)(14,29)(15,26)(16,27)(17,30)(19,33)(20,34)(22,39)(23,36)(24,37)(25,28)(31,40)(32,41)(35,38);(6,8,9,7)(11,19)(17,31)(18,33)(20,28)(25,34)(30,40)(32,38)(35,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16005);