Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×C5×#1338 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle, prime radical | C23×C5 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(27,28,29,30,31);(4,7)(5,9)(8,10)(11,12,14,19,16,17,22,24,15,18,23,25,13,20,21,26);(3,4,6,8)(5,10,9,7)(11,13,15,16)(12,17,18,20)(14,22,23,21)(19,26,25,24)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4597);