Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19071 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3553 | 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)(22,23,28,33)(24,34,37,46)(25,38,39,47)(26,32,27,31)(29,41,44,36)(30,42,45,40)(35,43,49,53)(48,50,52,51);(6,7)(8,12)(9,13)(10,18)(11,14)(15,19)(16,20)(17,21)(22,24)(23,29)(25,35)(26,41)(27,36)(28,37)(30,43)(31,46)(32,34)(33,44)(38,52)(39,49)(40,51)(42,50)(45,53)(47,48);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,25,28,39)(23,30,33,45)(24,35,37,49)(26,40,27,42)(29,43,44,53)(31,38,32,47)(34,48,46,52)(36,50,41,51);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(25,39)(30,45)(35,49)(38,47)(40,42)(43,53)(48,52)(50,51)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12059);