Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18290 |
| 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 ◁ C23×C4×C5 ◁ G | |
1034 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)(8,16)(9,19)(13,25)(14,23)(15,20)(17,31)(18,30)(21,34)(24,26)(27,37)(28,35)(29,33)(32,36);(6,7)(8,13)(9,20)(10,15)(11,14)(12,19)(16,26)(17,32)(18,27)(21,35)(22,28)(23,34)(24,33)(25,31)(29,37)(30,36);(9,21)(12,23)(17,29)(19,34)(20,35)(24,25)(31,33)(32,37);(6,8,10,18)(7,13,15,27)(9,17,19,33)(11,16,22,30)(12,24,20,32)(14,26,28,36)(21,29,34,31)(23,25,35,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11278);