Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3459 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1004 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)(6,7,11,15)(8,16,18,21)(9,14,10,13)(12,19,20,17)(22,23,26,29)(24,27,31,34)(25,28,32,35)(30,36,37,33);(6,8)(7,12)(9,17)(10,19)(11,18)(13,16)(14,21)(15,20)(22,24)(23,27)(25,30)(26,31)(28,33)(29,34)(32,37)(35,36);(22,25)(23,28)(24,30)(26,32)(27,33)(29,35)(31,37)(34,36);(6,9,11,10)(7,13,15,14)(8,17,18,19)(12,16,20,21)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11129);