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×#3515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
7142 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)(11,12,16,17)(13,14,18,21)(15,23,24,19)(20,26,25,22);(12,17)(14,21)(19,23)(22,26);(3,4)(5,8)(6,7)(9,10)(11,13,16,18)(12,14,17,21)(15,20,24,25)(19,22,23,26);(11,14)(12,18)(13,17)(15,22)(16,21)(19,25)(20,23)(24,26);(3,5,6,9)(4,8,7,10)(11,15)(12,19)(13,20)(14,22)(16,24)(17,23)(18,25)(21,26)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55767);