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×#17767 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
862 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)(6,7,11,14)(8,15,17,20)(9,13,10,18)(12,19,21,16)(22,23,26,31,27,29,25,30)(24,32,33,28,35,37,34,36);(6,8,11,17)(7,12,14,21)(9,16,10,19)(13,20,18,15)(22,24)(23,28)(25,33)(26,34)(27,35)(29,36)(30,37)(31,32);(6,7)(8,12)(9,18)(10,13)(11,14)(15,16)(17,21)(19,20)(23,29)(28,36)(30,31)(32,37);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,25,27,26)(23,30,29,31)(24,33,35,34)(28,37,36,32)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10755);