Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60389 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ G | |
4770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7,11,15)(8,16,19,29)(9,13,20,27)(10,22,24,14)(12,23,26,18)(17,30,31,37)(21,34,33,28)(25,35,36,32);(6,8)(7,12)(9,17)(10,23)(11,19)(13,25)(14,29)(15,26)(16,22)(18,24)(20,31)(21,35)(27,36)(28,37)(30,34)(32,33);(10,24)(14,22)(16,29)(18,23)(21,33)(28,34)(30,37)(32,35);(6,9)(7,13)(8,17)(10,21)(11,20)(12,25)(14,28)(15,27)(16,30)(18,32)(19,31)(22,34)(23,35)(24,33)(26,36)(29,37);(9,20)(13,27)(17,31)(21,33)(25,36)(28,34)(30,37)(32,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53377);