Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33569 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C5 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×Q8 ◁ G | |
1082 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)(8,18)(9,15)(10,23)(11,26)(12,17)(13,19)(14,22)(16,28)(20,33)(21,36)(24,30)(25,32)(27,29)(31,37)(34,35);(6,8)(7,13)(9,19)(10,24)(11,27)(12,20)(14,29)(15,33)(16,31)(17,30)(18,23)(21,28)(22,37)(25,35)(26,34)(32,36);(6,9,12,10)(7,14,17,32)(8,19,20,24)(11,25,28,22)(13,29,30,36)(15,16,23,26)(18,34,33,31)(21,37,27,35);(8,20)(13,30)(18,33)(19,24)(21,27)(29,36)(31,34)(35,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26557);