Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#12657 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C42 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#97 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
334 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)(6,7,8,9)(10,11)(12,22,25,37)(13,18)(14,19)(15,30)(16,21)(17,24,33,38)(20,31)(23,34,26,40)(27,41,39,32)(28,35)(29,36);(7,9)(10,12,15,26,14,24,28,39,16,25,31,23,29,38,13,27)(11,17,20,32,19,22,18,34,21,33,30,41,36,37,35,40);(10,13,14,15,16,28,29,31)(11,18,19,30,21,35,36,20)(12,23,38,39,25,26,24,27)(17,32,37,40,33,41,22,34)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5645);