Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C2×C5×#3518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
7118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 6 generators on 31 points:
(1,2)(27,28,29,30,31);(4,7)(8,10)(11,12,17,15)(13,20,14,18)(16,23,24,19)(21,25,22,26);(11,13,17,14)(12,18,15,20)(16,21,24,22)(19,25,23,26);(3,4)(5,8)(6,7)(9,10)(11,14,17,13)(12,18,15,20)(16,22,24,21)(19,25,23,26);(11,15,17,12)(13,18,14,20)(16,23,24,19)(21,25,22,26);(3,5,6,9)(4,8,7,10)(11,16)(12,19)(13,21)(14,22)(15,23)(17,24)(18,25)(20,26)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55779);