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×#3512 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
7238 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)(13,18)(14,19)(15,23)(16,17)(20,24)(21,22)(25,26);(3,4,6,8)(5,7,9,10)(12,17)(14,18)(20,23)(22,25);(3,5,6,9)(4,7,8,10)(11,13,16,19)(12,18,17,14)(15,21,24,26)(20,25,23,22);(11,14,16,18)(12,19,17,13)(15,22,24,25)(20,26,23,21);(11,15,16,24)(12,20,17,23)(13,21,19,26)(14,22,18,25)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55797);