Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2213 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
790 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);(4,7)(5,10)(6,8)(11,12)(13,18)(14,17)(15,16);(3,4,8,11,9,12,6,7)(5,10,16,18,14,17,13,15);(3,5)(4,10)(6,13)(7,15)(8,16)(9,14)(11,18)(12,17)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26580);