Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33555 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C5 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
1146 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)(8,18)(9,14)(10,15)(11,16)(12,17)(13,20)(19,32)(21,33)(22,28)(23,37)(24,30)(25,31)(26,34)(27,36)(29,35);(6,8)(7,13)(9,19)(10,23)(11,20)(12,21)(14,26)(15,29)(16,18)(17,27)(22,34)(24,35)(25,36)(28,32)(30,37)(31,33);(6,9,12,10)(7,14,17,15)(8,19,21,23)(11,24,25,22)(13,26,27,29)(16,28,31,30)(18,32,33,37)(20,35,36,34);(6,10,12,9)(7,15,17,14)(8,19,21,23)(11,24,25,22)(13,26,27,29)(16,30,31,28)(18,32,33,37)(20,34,36,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26543);