Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#1383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #106 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C24×C5 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4)(5,6)(8,13,14,15)(9,17)(10,19)(11,18)(16,22,20,21);(4,6)(7,8,10,15,11,16,19,22,12,14,9,13,18,20,17,21)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5550);