Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18282 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#1600 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,12)(8,16)(9,20)(13,25)(14,23)(15,21)(19,30)(22,31)(24,26)(27,34)(28,36)(35,37);(6,7,10,15)(8,13,16,26)(9,21,20,12)(11,14,17,27)(18,25,29,24)(19,28,30,37)(22,34,31,23)(32,35,33,36);(8,16)(9,22)(12,23)(13,26)(18,32)(19,30)(20,31)(21,34)(24,36)(25,35)(28,37)(29,33);(6,8,17,30)(7,13,27,37)(9,18,31,32)(10,16,11,19)(12,24,34,36)(14,28,15,26)(20,29,22,33)(21,25,23,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11270);