Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ G | |
4874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2,3,4,5)(7,11)(9,16)(12,18)(13,17)(15,20)(19,21);(6,7)(8,12)(9,17)(10,13)(11,16)(14,19)(15,21)(18,20);(9,16)(11,17)(15,20)(18,21);(6,8)(7,12)(9,15)(10,14)(11,18)(13,19)(16,20)(17,21);(8,14)(12,19)(15,20)(18,21)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53376);