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) = C2×C4 |
| direct product | C5×#8252 |
| 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 | C23×C5 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
414 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)(6,7,10,11)(8,9,12,13)(14,15,18,23,19,24,27,22)(16,17,21,28,26,29,25,20);(6,8,9,11,10,12,13,7)(14,16,18,21,19,26,27,25)(15,20,23,17,24,28,22,29);(14,17,27,20,19,29,18,28)(15,21,22,16,24,25,23,26)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1240);