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×#3507 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| prime radical | C2×C5×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
7130 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);(5,9)(7,10)(11,12)(13,20)(14,15)(16,17)(18,23)(19,24)(21,26)(22,25);(3,4,6,8)(5,7,9,10)(12,17)(14,19)(18,20)(21,25);(3,5,6,9)(4,7,8,10)(11,13,16,23)(12,18,17,20)(14,21,19,25)(15,22,24,26);(11,14,16,19)(12,15,17,24)(13,21,23,25)(18,22,20,26);(11,15)(12,19)(13,22)(14,17)(16,24)(18,25)(20,21)(23,26)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55765);