Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2169 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C23×C5 | C22×C8 |
| prime radical | C22×C5×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ G | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4)(5,6)(7,8,11,15,12,9,16,18,10,13,19,21,20,17,14,22);(4,6)(8,13)(9,17)(15,21)(18,22);(7,9,11,18,12,13,16,21,10,17,19,22,20,8,14,15)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26419);