Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C22×C5×#508 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | D8 |
| commutator, Frattini, cosocle | C8 | C25×C5 |
| socle | C23×C5 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2)(37,38,39,40,41);(3,4);(5,6)(7,16)(8,13)(9,21)(10,25)(11,15)(12,18)(14,23)(17,32)(19,33)(20,31)(22,27)(24,35)(26,30)(28,36)(29,34);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,27)(14,30)(15,29)(16,21)(18,23)(20,34)(24,36)(25,28)(31,32)(33,35);(5,8,11,10)(6,13,15,25)(7,17,19,26)(9,20,24,23)(12,27,29,28)(14,21,31,35)(16,32,33,30)(18,22,34,36)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55616);