Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×C8×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C8 | D4 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
554 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)(6,7,8,9,10,11,12,13);(14,15,17,20,18,16,21,19);(14,16)(15,18)(17,19)(20,21)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1178);