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×#17682 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3515 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
876 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)(8,14,16,21)(9,13)(10,12)(11,15,19,17)(18,20)(22,23,27,31,28,24,26,30)(25,32,33,37,35,29,34,36);(6,8)(7,11)(9,17)(10,16)(12,19)(13,21)(14,20)(15,18)(22,24,28,23)(25,32,35,29)(26,31,27,30)(33,37,34,36);(7,12)(11,19)(13,20)(14,21)(22,25)(23,29)(24,32)(26,34)(27,33)(28,35)(30,37)(31,36);(22,26,28,27)(23,30,24,31)(25,33,35,34)(29,36,32,37)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10670);