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×#18283 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#538 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(11,17)(18,21)(22,23,26,28)(24,30,32,37)(25,33,35,29)(27,34,36,31);(6,7)(8,11)(9,12)(10,13)(14,18)(15,17)(16,19)(20,21)(22,24,26,32)(23,27,28,36)(25,34,35,31)(29,37,33,30);(6,8)(7,11)(9,14)(10,15)(12,18)(13,17)(16,20)(19,21)(22,23)(24,27)(25,33)(26,28)(29,35)(30,34)(31,37)(32,36);(6,9,10,16)(7,12,13,19)(8,14,15,20)(11,18,17,21)(23,28)(27,36)(29,33)(30,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11271);