Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19575 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(6,7,11,14)(8,15,17,20)(9,13,10,19)(12,18,21,16)(22,23,27,31)(24,28,33,42)(25,35,37,45)(26,38,40,30)(29,39,44,36)(32,46,47,53)(34,50,49,43)(41,51,52,48);(6,8,10,16,11,17,9,18)(7,12,13,15,14,21,19,20)(22,24)(23,28)(25,32)(26,34)(27,33)(29,41)(30,43)(31,42)(35,46)(36,48)(37,47)(38,50)(39,51)(40,49)(44,52)(45,53);(6,7)(8,12)(9,19)(10,13)(11,14)(15,16)(17,21)(18,20)(22,25,27,37)(23,29,31,44)(24,32,33,47)(26,39,40,36)(28,41,42,52)(30,45,38,35)(34,51,49,48)(43,53,50,46);(6,9,11,10)(7,13,14,19)(8,16,17,18)(12,20,21,15)(24,33)(28,42)(32,47)(34,49)(41,52)(43,50)(46,53)(48,51)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12563);