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×#17627 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×D4 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14,16,11)(9,13)(10,12)(15,20,21,18)(17,19)(22,23)(24,30)(25,33)(26,28)(27,31)(29,35)(32,37)(34,36);(6,8,10,16)(7,11,12,14)(9,15,17,21)(13,18,19,20)(22,24,26,32)(23,27,28,36)(25,34,35,31)(29,37,33,30);(7,12)(9,17)(11,14)(15,21)(23,28)(27,36)(29,33)(30,37);(6,9,10,17)(7,13,12,19)(8,15,16,21)(11,18,14,20)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10615);