Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18287 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C25×C5 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1064 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)(6,7,10,15)(8,16,19,29)(9,20,23,14)(11,13,24,28)(12,21,26,17)(18,30,32,37)(22,33,35,27)(25,34,36,31);(6,8,10,19)(7,12,15,26)(9,21,23,17)(11,18,24,32)(13,25,28,36)(14,29,20,16)(22,34,35,31)(27,37,33,30);(9,22)(14,27)(16,30)(17,31)(20,33)(21,34)(23,35)(29,37);(7,13)(12,25)(14,27)(15,28)(16,30)(20,33)(26,36)(29,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11275);