Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33768 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,20,25)(16,21,27,42)(17,30,33,64)(18,34,38,63)(19,39,36,24)(22,45,47,31)(23,48,50,37)(26,52,53,75)(28,58,56,74)(29,61,57,44)(32,65,40,49)(35,51,66,46)(41,67,68,54)(43,71,70,60)(55,76,62,72)(59,73,77,69);(6,8)(7,10)(9,13)(11,12)(14,16,20,27)(15,21,25,42)(17,26,33,53)(18,28,38,56)(19,29,36,57)(22,41,47,68)(23,43,50,70)(24,44,39,61)(30,52,64,75)(31,54,45,67)(32,55,40,62)(34,58,63,74)(35,59,66,77)(37,60,48,71)(46,69,51,73)(49,72,65,76);(8,13)(10,12)(14,17,20,33)(15,22,25,47)(16,26,27,53)(18,35,38,66)(19,40,36,32)(21,41,42,68)(23,49,50,65)(24,51,39,46)(28,59,56,77)(29,62,57,55)(30,63,64,34)(31,48,45,37)(43,72,70,76)(44,73,61,69)(52,74,75,58)(54,71,67,60);(7,11)(10,12)(16,27)(21,42)(26,53)(28,56)(29,57)(41,68)(43,70)(44,61)(52,75)(54,67)(55,62)(58,74)(59,77)(60,71)(69,73)(72,76)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26756);