Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C5×#62800 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| socle | C22×C5 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
7856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 21 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(15,18)(16,20);(7,10)(11,13);(14,15)(16,19)(17,18)(20,21);(15,18)(19,21);(14,16,17,20)(15,19,18,21);(6,8,9,12)(7,11,10,13)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55788);