Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C5×#1930 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
1026 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);(3,4,6,5)(8,12)(9,15)(10,17)(13,20)(14,19)(18,21);(3,5,6,4)(7,8,10,12,11,14,17,19)(9,13,16,18,15,21,22,20);(7,9,10,16,11,15,17,22)(8,13,12,18,14,21,19,20)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14061);