Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#459 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
706 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)(9,20)(10,21)(13,27)(14,25)(15,31)(17,22)(18,33)(19,34)(23,35)(24,29)(28,32)(30,37);(6,7,9,12,11,14,20,25)(8,13,17,24,19,28,18,30)(10,15,21,26,23,31,35,36)(16,29,33,27,34,37,22,32);(6,8)(7,13)(9,17)(10,22)(11,19)(12,24)(14,28)(15,32)(16,21)(18,20)(23,33)(25,30)(26,29)(27,31)(34,35)(36,37);(7,14)(12,25)(13,28)(15,31)(24,30)(26,36)(27,32)(29,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26583);