Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C5×#1975 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C4×D4 |
| prime radical | C22×C5×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C23×C5×C8 ◁ G | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(4,6)(7,8,11,15,12,16,10,13)(9,17,20,22,18,21,19,14);(3,4,5,6)(8,13,16,15)(9,18)(14,17,22,21)(19,20);(7,9,10,19,12,18,11,20)(8,14,15,17,16,22,13,21)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14227);