Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8251 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,11)(8,9)(12,13)(14,15,18,24,20,25,19,23)(16,26,27,29,28,17,22,21);(6,7,9,12,10,13,8,11)(14,16,19,22,20,28,18,27)(15,21,23,17,25,29,24,26);(14,17,18,21,20,26,19,29)(15,22,24,16,25,27,23,28)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1239);