Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C5 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×C5×#340 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C5 | D4 |
| commutator | C4 | C43×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle, prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C5 ◁ C22×C42×C5 ◁ G | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4)(21,22,23,24,25);(5,6)(7,8)(9,10,11,12)(14,18)(15,16)(19,20);(6,8)(13,14,16,19,17,20,15,18)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3690);