Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1790 |
| commutator | C2×Q8 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3551 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,18)(9,22,10,15)(11,16)(12,17)(13,19,30,24)(14,23,31,25)(20,34)(21,32)(26,28)(27,33)(29,35,37,36);(6,8)(7,13)(9,19)(10,24)(11,26)(12,21)(14,28)(15,32)(16,29)(17,30)(18,22)(20,27)(23,36)(25,35)(31,34)(33,37);(6,9,12,10)(7,14,17,31)(8,19,21,24)(11,25,27,23)(13,28,30,34)(15,33,22,16)(18,29,32,37)(20,36,26,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5768);