Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11306 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #400 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
530 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)(10,11,16,19,15,20,14,21)(12,22,26,32,25,33,24,17)(13,18,29,35,30,36,28,37)(23,34,27,40,39,41,38,31);(6,7)(8,9)(10,12,16,26,15,25,14,24)(11,17,21,33,20,32,19,22)(13,27,28,23,30,38,29,39)(18,34,35,40,36,41,37,31);(7,9)(10,13,14,28,15,30,16,29)(11,18,19,35,20,36,21,37)(12,23,24,38,25,39,26,27)(17,31,22,34,32,40,33,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4294);