Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33571 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C5 |
| prime radical | C5×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
1084 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)(9,15)(10,24)(11,14)(12,17)(13,20)(16,27)(19,30)(21,34)(22,33)(23,31)(25,32)(26,35)(28,37)(29,36);(6,8,12,22)(7,13,17,30)(9,20,10,19)(11,26,27,21)(14,28,16,29)(15,33,24,18)(23,36,25,37)(31,35,32,34);(7,14)(11,27)(13,28)(15,31)(16,17)(18,34)(21,26)(23,25)(24,32)(29,30)(33,35)(36,37);(6,9,12,10)(7,15,17,24)(8,19,22,20)(11,23,27,25)(13,18,30,33)(14,31,16,32)(21,36,26,37)(28,34,29,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26559);