Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
958 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)(6,7,10,15,11,16,26,14)(8,17,20,28,21,30,19,12)(9,22,24,29,25,31,23,13)(18,27,33,36,34,37,35,32);(6,8,10,20,11,21,26,19)(7,12,15,17,16,28,14,30)(9,18,24,33,25,34,23,35)(13,27,22,36,29,37,31,32);(6,9,11,25)(7,13,16,29)(8,18,21,34)(10,23,26,24)(12,27,28,37)(14,22,15,31)(17,32,30,36)(19,33,20,35);(6,7)(8,12)(9,13)(10,15)(11,16)(14,26)(17,20)(18,27)(19,30)(21,28)(22,24)(23,31)(25,29)(32,35)(33,36)(34,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18186);