Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12767 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1790 |
| commutator | C2×D4 | C22×C4×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| prime radical | C5×#3543 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×D4 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5)(6,7)(8,15,17,12)(9,13,19,20)(10,14)(11,16)(18,21);(6,8)(7,11)(9,16)(10,17)(12,20)(13,15)(14,21)(18,19);(6,8)(7,12)(9,18)(10,17)(11,20)(13,21)(14,15)(16,19)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5755);