Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#19627 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
784 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)(8,14)(9,17)(10,12)(11,16)(13,18)(15,20)(19,21)(22,23)(24,30)(25,27)(26,34)(28,32)(29,33)(31,36)(35,37);(6,8,9,16,10,15,18,21)(7,11,13,14,12,19,17,20)(23,27)(28,35)(29,34)(30,36);(7,12)(11,19)(13,17)(14,20)(22,24,26,32,25,31,33,37)(23,28,29,30,27,35,34,36);(8,15)(11,19)(14,20)(16,21)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12615);