Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23414 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1980 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)(8,16)(9,13)(10,22)(11,15)(12,18)(14,24)(17,32)(19,30)(20,28)(21,36)(23,26)(25,37)(27,34)(29,35)(31,33);(6,8)(7,12)(9,17)(10,23)(11,19)(13,27)(14,30)(15,26)(16,22)(18,24)(20,33)(21,37)(25,28)(29,31)(32,36)(34,35);(6,9)(7,13)(8,17)(10,21)(11,20)(12,25)(14,29)(15,28)(16,31)(18,34)(19,33)(22,36)(23,37)(24,35)(26,27)(30,32);(9,20)(13,28)(17,33)(21,35)(25,27)(29,36)(31,32)(34,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16402);