Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
810 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)(8,15)(9,13)(10,18)(11,14)(12,16)(17,20)(19,21)(22,23,28,31)(24,34,37,46)(25,38,40,47)(26,33,27,32)(29,41,44,36)(30,42,45,39)(35,43,51,53)(48,49,52,50);(6,7)(8,12)(9,13)(10,18)(11,14)(15,19)(16,20)(17,21)(22,24,28,37)(23,29,31,44)(25,35,40,51)(26,41,27,36)(30,43,45,53)(32,46,33,34)(38,52,47,48)(39,50,42,49);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,25)(23,30)(24,35)(26,39)(27,42)(28,40)(29,43)(31,45)(32,38)(33,47)(34,48)(36,49)(37,51)(41,50)(44,53)(46,52);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(23,31)(29,44)(30,45)(32,33)(34,46)(38,47)(43,53)(48,52)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12001);