Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12934 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1853 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
394 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)(6,7,10,17,12,18,11,16)(8,19,22,31,23,34,21,14)(9,24,26,33,28,32,27,15)(13,25,29,20,30,36,35,37);(6,8)(7,13,18,30)(9,25)(10,21)(11,22)(12,23)(14,15,31,33)(16,29,17,35)(19,32,34,24)(20,27)(26,37)(28,36);(6,8)(7,14)(9,20)(10,22)(11,21)(12,23)(13,15)(16,34)(17,19)(18,31)(24,29)(25,27)(26,36)(28,37)(30,33)(32,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5922);