Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#8144 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C5 | C23 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C23×C5 | C2×C42 |
| prime radical | C24×C5 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C5 ◁ G | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,16,13,12,15,17);(7,9)(11,12)(16,17)(18,19,20,22,21,23,24,25);(10,12,14,17,13,11,15,16)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1132);