Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1074 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)(6,7,12,16)(8,17,19,32)(9,14,22,31)(10,23,25,34)(11,15,21,30)(13,26,29,18)(20,27,33,37)(24,28,35,36);(6,8)(7,13)(9,18)(10,24)(11,20)(12,19)(14,27)(15,28)(16,29)(17,23)(21,33)(22,26)(25,35)(30,36)(31,37)(32,34);(6,9,21,25)(7,14,30,34)(8,18,33,35)(10,12,22,11)(13,27,36,32)(15,23,16,31)(17,29,37,28)(19,26,20,24);(7,15)(13,28)(14,23)(16,30)(17,27)(29,36)(31,34)(32,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11137);