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×#17632 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C23×C5 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
874 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)(9,17)(10,12)(11,15)(13,19)(16,21)(18,20)(23,28,30,34)(29,35,37,36);(6,8,10,16)(7,11,12,20)(9,18,19,15)(13,21,17,14)(22,23)(24,29)(25,28)(26,34)(27,30)(31,36)(32,37)(33,35);(7,12)(11,20)(13,17)(14,21)(22,24,27,32)(23,29,30,37)(25,33,26,31)(28,35,34,36);(22,25,27,26)(23,28,30,34)(24,31,32,33)(29,36,37,35)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10620);