Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| 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 | |
618 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)(8,14,16,21)(9,18,15,12)(11,19,20,17)(22,23,26,30)(24,27,31,35)(25,28,33,29)(32,34,37,36);(6,8,15,19,10,16,9,17)(7,11,18,21,13,20,12,14)(22,24)(23,27)(25,32)(26,31)(28,34)(29,36)(30,35)(33,37);(8,16)(11,20)(14,21)(17,19)(23,28)(25,33)(27,34)(29,30)(32,37)(35,36);(24,31)(27,35)(32,37)(34,36)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12417);