Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23898 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1718 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)(6,7)(8,12)(9,19)(10,22)(11,15)(13,21)(14,24)(16,31)(17,34)(18,27)(20,30)(23,29)(25,33)(26,36)(28,35)(32,37);(6,8,11,18)(7,12,15,27)(9,16,20,32)(10,17,24,36)(13,28,29,25)(14,26,22,34)(19,37,30,31)(21,33,23,35);(6,9)(7,13)(8,16)(10,23)(11,20)(12,25)(14,30)(15,29)(17,35)(18,32)(19,22)(21,24)(26,37)(27,28)(31,34)(33,36);(10,24)(14,22)(17,36)(19,30)(21,23)(26,34)(31,37)(33,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16886);