Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12715 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1781 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×2−1+4 ◁ G | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17,20,30)(9,14,21,12)(10,15)(11,16)(13,19,28,23)(18,33)(22,29,36,26)(24,31)(25,35)(27,37)(32,34);(7,12)(9,21)(13,25)(14,16)(15,26)(17,32)(18,34)(22,36)(27,28)(29,31)(30,33)(35,37);(6,8)(7,13)(9,18)(10,23)(11,20)(12,25)(14,27)(15,30)(16,28)(17,31)(19,24)(21,34)(22,37)(26,33)(29,32)(35,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5703);