Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#8852 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C22×C5×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5)(10,13)(14,15,18,16,17,19,21,20);(6,7)(8,10)(9,11)(12,13)(14,16,18,19,17,20,21,15);(6,8)(7,10)(9,12)(11,13)(15,19)(16,20)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1840);