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×#18297 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1084 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)(6,7,11,16)(8,17,20,26)(9,14,10,23)(12,15,21,29)(13,24,28,18)(19,31,32,37)(22,30,25,34)(27,35,36,33);(6,8)(7,13)(9,18)(10,24)(11,20)(12,19)(14,26)(15,27)(16,28)(17,23)(21,32)(22,33)(25,35)(29,36)(30,37)(31,34);(6,9,21,25)(7,14,29,34)(8,18,32,35)(10,12,22,11)(13,26,36,31)(15,30,16,23)(17,27,37,28)(19,33,20,24);(7,15)(13,27)(14,30)(16,29)(17,31)(23,34)(26,37)(28,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11285);