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×#3521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
7314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 31 points:
(1,2)(27,28,29,30,31);(4,7)(8,10)(11,12)(13,19)(14,22)(15,16)(17,21)(18,23)(20,24)(25,26);(12,16)(17,19)(18,22)(24,25);(3,4)(5,8)(6,7)(9,10)(11,13,15,21)(12,17,16,19)(14,20,23,26)(18,24,22,25);(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,20)(15,23)(16,22)(17,24)(19,25)(21,26)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55768);