Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17727 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
940 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)(6,7)(8,17)(9,14)(10,15)(11,20)(12,16)(13,18)(19,21)(22,23,26,30,27,31,25,29)(24,32,33,28,35,36,34,37);(6,8,12,9)(7,13,16,21)(10,18,11,19)(14,20,17,15)(22,24)(23,28)(25,33)(26,34)(27,35)(29,36)(30,32)(31,37);(6,9,12,8)(7,14,16,17)(10,18,11,19)(13,20,21,15)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32);(6,10,12,11)(7,15,16,20)(8,18,9,19)(13,14,21,17)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10715);