Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3552 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,17)(15,20)(16,18)(19,21)(22,23,27,24)(25,32,33,37)(26,31,36,29)(28,35,30,34);(7,11)(9,17)(12,18)(13,14)(16,21)(19,20)(22,24,27,23)(25,30,33,28)(26,29,36,31)(32,35,37,34);(6,8)(7,12)(9,16)(10,15)(11,18)(13,20)(14,19)(17,21)(22,25)(23,28)(24,30)(26,35)(27,33)(29,37)(31,32)(34,36);(7,13)(8,15)(11,14)(16,21)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11721);