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×#17761 |
| 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 | |
896 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)(8,15)(9,11)(10,14)(12,16)(13,18)(17,20)(19,21)(22,23,26,31,27,29,25,30)(24,32,34,28,35,37,33,36);(7,11)(9,18)(12,19)(13,14)(15,20)(17,21)(22,24,27,35)(23,28,29,36)(25,34,26,33)(30,37,31,32);(6,8)(7,12)(9,17)(10,16)(11,19)(13,20)(14,15)(18,21)(23,29)(28,36)(30,31)(32,37);(8,16)(12,15)(17,21)(19,20)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10749);