Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,18,23,19,22,24,36)(16,20,27,32,26,33,30,37)(17,21,29,35,28,34,38,44)(25,31,40,43,39,42,41,45);(14,16,24,27,19,26,18,30)(15,20,23,37,22,33,36,32)(17,25,38,40,28,39,29,41)(21,31,35,45,34,42,44,43);(7,10)(11,13)(14,17,19,28)(15,21,22,34)(16,25,26,39)(18,29,24,38)(20,31,33,42)(23,35,36,44)(27,40,30,41)(32,43,37,45);(6,8,9,12)(7,11,10,13)(17,28)(21,34)(25,39)(29,38)(31,42)(35,44)(40,41)(43,45)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14417);