Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#22257 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C23×C5 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(6,7)(8,15)(9,13)(10,19)(11,14)(12,17)(16,21)(18,20)(22,23,27,31,28,32,26,41)(24,29,35,43,36,44,34,50)(25,37,40,45,38,51,39,30)(33,46,49,52,47,53,48,42);(6,8)(7,12)(9,18)(10,16)(11,17)(13,21)(14,15)(19,20)(22,24,27,35,28,36,26,34)(23,29,31,43,32,44,41,50)(25,33,40,49,38,47,39,48)(30,42,37,46,45,52,51,53);(6,9,11,10)(7,13,14,19)(8,16,17,18)(12,20,15,21)(22,25,26,39,28,38,27,40)(23,30,31,37,32,45,41,51)(24,33,34,48,36,47,35,49)(29,42,43,46,44,52,50,53);(6,10,11,9)(7,13,14,19)(8,16,17,18)(12,21,15,20)(25,38)(30,45)(33,47)(37,51)(39,40)(42,52)(46,53)(48,49)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15245);