Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23991 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3027 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,13)(9,19)(10,12)(11,23)(14,20)(15,22)(16,26)(17,27)(18,25)(21,29)(24,28)(30,36)(31,32)(33,34)(35,37);(7,12)(8,16,18,24)(13,26,25,28)(14,19)(15,23)(17,30,31,35)(27,36,32,37)(29,33);(6,8)(7,13)(9,17)(10,18)(11,24)(12,25)(14,27)(15,26)(16,22)(19,32)(20,31)(21,35)(23,28)(29,36)(30,34)(33,37);(6,9)(7,14)(8,17)(10,20)(11,21)(12,19)(13,27)(15,29)(16,30)(18,31)(22,34)(23,33)(24,35)(25,32)(26,36)(28,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16979);