Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12766 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | #1790 |
| prime radical | C5×#3529 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,15,26,33)(8,16,19,34,17,30,24,36)(9,20,23,32,22,13,25,31)(12,27,29,21,28,35,37,18);(6,8)(7,12)(9,21)(10,24)(11,17)(13,30)(14,29)(15,28)(16,20)(18,22)(19,26)(23,27)(25,35)(31,36)(32,34)(33,37);(8,17)(12,28)(13,20)(18,21)(19,24)(27,35)(29,37)(31,32)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5754);