Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(6,7,10,14,11,9,20,13)(8,15,18,16,19,21,17,12);(6,8,10,18,11,19,20,17)(7,12,14,15,9,16,13,21);(6,9)(7,11)(8,16)(10,14)(12,19)(13,20)(15,18)(17,21);(6,9)(7,11)(8,16)(10,13)(12,19)(14,20)(15,17)(18,21)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18184);