Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #481 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
318 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)(7,8)(10,11)(12,20,23,36)(13,16)(14,26)(15,29)(17,22,31,38)(18,28)(19,21)(24,39,30,34)(25,40)(27,33,41,35)(32,37);(6,7,9,8)(11,16)(14,25)(17,31)(19,29)(22,38)(27,41)(28,37)(33,35);(6,8,9,7)(10,12,21,24,13,23,15,30)(11,17,29,33,16,31,19,35)(14,27,37,38,25,41,28,22)(18,34,40,36,32,39,26,20)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6127);