Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18414 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1042 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)(9,19)(13,25)(14,23)(15,20)(17,29)(18,32)(21,34)(24,27)(26,36)(28,35)(30,31)(33,37);(6,7)(8,13)(9,20)(10,15)(11,14)(12,19)(16,27)(17,26)(18,33)(21,35)(22,28)(23,34)(24,31)(25,32)(29,37)(30,36);(8,17)(9,21)(12,23)(13,26)(16,29)(18,32)(19,34)(20,35)(24,36)(25,33)(27,37)(30,31);(6,8)(7,13)(9,18)(10,17)(11,16)(12,24)(14,27)(15,26)(19,31)(20,33)(21,30)(22,29)(23,25)(28,37)(32,34)(35,36)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11402);