Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3109 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
660 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)(6,7,10,13)(8,14,15,21)(9,17,18,12)(11,19,20,16)(22,23)(24,27)(25,28)(26,30)(29,33)(31,35)(32,34)(36,37);(6,8,9,16,10,15,18,19)(7,11,12,14,13,20,17,21)(22,24)(23,27)(25,32)(26,31)(28,34)(29,36)(30,35)(33,37);(23,28)(25,33)(27,34)(29,30)(32,37)(35,36);(8,15)(11,20)(14,21)(16,19)(24,31)(27,35)(32,37)(34,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12445);