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×#33568 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C5 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×Q8 ◁ G | |
1110 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)(8,17)(9,21)(10,13)(11,16)(12,19)(14,24)(15,26)(18,32)(20,34)(22,29)(23,31)(25,35)(27,33)(28,36)(30,37);(6,8)(7,12)(9,22)(10,25)(11,18)(13,28)(14,32)(15,30)(16,29)(17,21)(19,24)(20,26)(23,36)(27,37)(31,34)(33,35);(7,13)(10,26)(12,28)(14,33)(15,16)(17,34)(20,25)(21,31)(23,27)(29,30)(32,35)(36,37);(8,18)(12,29)(17,32)(19,22)(20,25)(28,30)(34,35)(36,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26556);