Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33854 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#443 ◁ G | |
656 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)(7,13)(8,18)(9,21)(10,11)(14,31)(15,33)(16,35)(17,30)(19,36)(20,26)(22,27)(24,25)(28,32)(29,34);(6,7)(8,14)(9,22)(10,16)(11,27)(12,17)(13,21)(15,19)(18,32)(20,33)(23,28)(24,30)(25,35)(26,29)(31,36)(34,37);(6,8)(7,14)(9,19)(10,26)(11,18)(12,20)(13,28)(15,22)(16,29)(17,33)(21,23)(24,37)(25,36)(27,32)(30,34)(31,35);(6,8)(7,15)(9,23)(10,18)(11,26)(12,20)(13,29)(14,22)(16,28)(17,31)(19,21)(24,36)(25,37)(27,34)(30,32)(33,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26842);