Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×C5×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C23×C5 | C22⋊C8 |
| prime radical | C22×C5×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ C23×C5×C8 ◁ G | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2)(37,38,39,40,41);(3,4);(5,6,9,14,10,15,24,32,11,16,25,33,8,13,22,30)(7,17,19,34,18,28,35,31,20,29,23,36,21,12,26,27);(5,7)(6,12)(8,21)(9,23)(10,18)(11,20)(13,29)(14,31)(15,17)(16,28)(19,25)(22,35)(24,26)(27,33)(30,34)(32,36)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26326);