Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17768 |
| 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 | |
918 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)(6,7,9,13,10,11,16,18)(8,12,15,19,14,17,20,21)(22,23)(24,31)(25,28)(26,30)(27,35)(29,33)(32,36)(34,37);(7,11)(9,16)(12,17)(15,20)(22,24)(23,29)(25,32)(26,33)(27,36)(28,34)(30,37)(31,35);(6,8,10,14)(7,12,11,17)(9,15,16,20)(13,19,18,21)(22,25,28,23)(24,32,34,29)(26,30,27,35)(31,33,37,36);(8,14)(12,17)(15,20)(19,21)(22,26,28,27)(23,30,25,35)(24,33,34,36)(29,37,32,31)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10756);