Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8440 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C5×#3481 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
884 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)(6,7,10,12,9,8,13,11)(15,18,19,20);(10,13)(11,12)(14,15)(16,20)(17,19)(18,21);(6,8)(7,9)(10,11)(12,13)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1428);