Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19007 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,10,12)(8,14,16,21)(9,17,19,13)(11,18,20,15)(22,23,27,24)(25,32,33,37)(26,31,36,29)(28,35,30,34);(6,8)(7,11)(9,18)(10,16)(12,20)(13,21)(14,17)(15,19)(22,24,27,23)(25,30,33,28)(26,29,36,31)(32,35,37,34);(9,19)(13,17)(14,21)(15,18)(22,25)(23,28)(24,30)(26,35)(27,33)(29,37)(31,32)(34,36);(7,12)(11,20)(13,17)(14,21)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11995);