Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10690 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #401 |
| commutator, Frattini, cosocle | C25 | C23×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C25×C5 ◁ G | |
372 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)(8,17,20,28)(9,21,36,30)(10,14)(11,16)(12,19,27,35)(13,23,31,24)(15,25)(18,32,33,29)(22,26,34,37);(6,8)(7,12,16,28)(9,22,23,33)(10,20)(11,19)(13,29,31,37)(14,27,15,17)(18,24,34,36)(21,32,30,26)(25,35);(6,9)(7,13,14,30)(8,18,19,33)(10,23)(11,24)(12,26,17,32)(15,31,16,21)(20,34,35,22)(25,36)(27,29,28,37)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3678);