Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C8 |
| direct product | C5×#8270 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C2×D4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C4⋊C4 |
| prime radical | C23×C5 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(7,11,13,16)(12,18,19,20)(22,23)(24,27)(25,29)(26,28)(30,34)(31,33)(32,35)(36,37);(6,7)(8,12)(9,16)(10,13)(11,17)(14,20)(15,19)(18,21)(22,24,25,30,26,31,32,36)(23,27,29,34,28,33,35,37);(6,8,9,14,10,15,17,21)(7,12,16,20,13,19,11,18)(23,28)(27,33)(29,35)(34,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1258);