Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C8×C16 · Schur multiplier (H2) = C4 |
| direct product | C5×#7329 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C16 | C22 |
| commutator | C2 | C5×C8×C16 |
| Frattini, cosocle | C4×C16 | C22×C5 |
| socle, prime radical | C22×C5 | C4×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C16 ◁ G | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5)(14,15,16,21,17,22,24,33,18,23,25,34,27,35,36,43,19,20,26,30,28,31,37,40,29,32,38,41,39,42,44,45);(6,7,8,10,9,11,12,13)(15,20)(21,30)(22,31)(23,32)(33,40)(34,41)(35,42)(43,45)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 317);