Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17812 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C4×C5×#41 ◁ G | |
602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,11)(9,12)(13,17)(14,18)(16,19)(20,21)(22,23,25,28,26,29,32,35)(24,27,31,34,30,33,36,37);(6,7,12,18,10,14,9,11)(8,13,19,21,15,20,16,17)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(6,8)(7,13)(9,16)(10,15)(11,17)(12,19)(14,20)(18,21)(24,30)(27,33)(31,36)(34,37);(8,15)(13,20)(16,19)(17,21)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10800);