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×#25197 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
968 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)(6,7,11,15,12,16,10,24)(8,17,19,13,20,28,25,29)(9,21,22,31,23,30,26,14)(18,33,35,37,36,27,34,32);(6,8,11,19,12,20,10,25)(7,13,15,28,16,29,24,17)(9,18,22,35,23,36,26,34)(14,27,21,32,31,33,30,37);(6,9)(7,14)(8,18)(10,22)(11,26)(12,23)(13,27)(15,30)(16,31)(17,32)(19,34)(20,36)(21,24)(25,35)(28,37)(29,33);(6,7,12,16)(8,13,20,29)(9,14,23,31)(10,24,11,15)(17,19,28,25)(18,27,36,33)(21,26,30,22)(32,34,37,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18185);