Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C16 · Schur multiplier (H2) = C23 |
| direct product | C5×#7199 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C22 | C4×C5×C16 |
| Frattini, cosocle | C23×C8 | C22×C5 |
| socle | C22×C5 | #336 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C23×C5×C8 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,17,22,19,16,21,25,20,24,18,23,28,27,29,26);(6,8)(7,10)(9,13)(11,12)(14,16,18,26,19,24,29,22,20,27,17,25,28,15,21,23)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 187);