Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#10952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
890 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)(7,9)(10,11)(12,17,23,32)(13,25)(14,21)(15,20)(16,19)(18,27)(22,36,38,39)(24,35,29,31)(26,33)(28,34)(30,40,41,37);(10,12,15,24,16,23,14,29)(11,17,21,31,19,32,20,35)(13,22,28,37,26,38,27,40)(18,30,33,36,34,41,25,39);(6,7,8,9)(10,13,14,27,16,26,15,28)(11,18,20,25,19,34,21,33)(12,22,29,40,23,38,24,37)(17,30,35,39,32,41,31,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3940);