Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33550 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C5 |
| socle | C2×C5 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
1144 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)(8,16)(9,13)(10,14)(11,15)(12,19)(17,28)(18,29)(20,25)(21,26)(22,27)(23,31)(24,32)(30,36)(33,35)(34,37);(6,8)(7,12)(9,18)(10,19)(11,17)(13,24)(14,16)(15,23)(20,32)(21,30)(22,31)(25,29)(26,34)(27,28)(33,37)(35,36);(8,17)(10,22)(12,23)(16,28)(18,30)(20,33)(24,34)(29,36);(6,9,11,21)(7,13,15,26)(8,18,17,30)(10,20,22,33)(12,24,23,34)(14,25,27,35)(16,29,28,36)(19,32,31,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26538);