Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3214 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,13,9,19)(8,15,18,20,17,21,16,12)(22,23,27,29,26,25,31,34)(24,30,32,37,33,28,35,36);(6,8)(7,12,13,20)(9,16)(10,18)(11,17)(14,15,19,21)(22,24,31,32,26,33,27,35)(23,28,29,37,25,30,34,36);(7,13)(9,10)(12,20)(16,18)(22,25)(23,26)(24,30)(27,29)(28,33)(31,34)(32,36)(35,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2359);