Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#2165 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | #432 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,5,10,6,11,13,18,7,8,12,15,14,9,16,17);(4,8)(5,12)(9,11)(13,16);(4,9,8,11)(5,12)(10,18,15,17)(13,16)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26412);