Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18248 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
992 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)(7,12)(9,20)(13,24)(14,23)(15,21)(18,28)(22,30)(25,31)(26,34)(27,35)(32,33)(36,37);(6,7)(8,13)(9,21)(10,15)(11,14)(12,20)(16,25)(17,26)(18,31)(19,27)(22,34)(23,30)(24,28)(29,37)(32,35)(33,36);(8,16)(9,22)(12,23)(13,25)(18,32)(19,29)(20,30)(21,34)(24,36)(27,37)(28,33)(31,35);(6,8,17,29)(7,13,26,37)(9,18,30,32)(10,16,11,19)(12,24,34,36)(14,27,15,25)(20,28,22,33)(21,31,23,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11236);