Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C5×#62826 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
6670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 6 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,14)(10,12)(11,15)(13,20)(17,29)(18,25)(19,30)(21,28)(22,24)(23,32)(26,35)(27,33)(31,37)(34,36);(7,12)(8,17)(9,21)(14,27)(15,25)(19,34)(20,32)(22,33)(24,28)(26,30)(31,35)(36,37);(6,8,18,32)(7,13,25,29)(9,19,33,34)(10,17,11,20)(12,23,15,16)(14,26,27,37)(21,35,22,31)(24,36,28,30);(7,12)(13,23)(14,24)(15,25)(16,29)(26,36)(27,28)(30,37);(6,9,10,22)(7,14,12,24)(8,19,17,31)(11,21,18,33)(13,26,23,36)(15,28,25,27)(16,30,29,37)(20,35,32,34);(7,15)(12,25)(13,16)(14,28)(23,29)(24,27)(26,30)(36,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55814);