Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17680 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3515 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14,16,21)(9,13)(10,12)(11,15,19,17)(18,20)(22,23,27,30,28,24,26,32)(25,31,35,37,33,29,34,36);(6,8)(7,11)(9,17)(10,16)(12,19)(13,21)(14,20)(15,18)(22,24,28,23)(25,31,33,29)(26,30,27,32)(34,36,35,37);(7,12)(11,19)(13,20)(14,21)(22,25)(23,29)(24,31)(26,34)(27,35)(28,33)(30,36)(32,37);(25,33)(29,31)(34,35)(36,37)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10668);