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×#33572 |
| 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 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
1108 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)(8,19)(9,17)(10,16)(11,26)(12,28)(13,18)(14,20)(15,24)(21,29)(22,36)(23,34)(25,33)(27,32)(30,35)(31,37);(6,8)(7,14)(9,20)(10,22)(11,27)(12,21)(13,23)(15,29)(16,31)(17,34)(18,32)(19,26)(24,37)(25,35)(28,30)(33,36);(6,9,13,11)(7,15,18,33)(8,20,23,27)(10,24,12,25)(14,29,32,36)(16,26,28,17)(19,30,34,31)(21,35,22,37);(6,10,13,12)(7,16,18,28)(8,21,23,22)(9,24,11,25)(14,30,32,31)(15,26,33,17)(19,29,34,36)(20,35,27,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26560);