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×#3516 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| 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);(4,7)(8,10)(12,15)(13,19)(18,22)(20,25);(11,12,16,15)(13,17,19,21)(14,22,23,18)(20,26,25,24);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,20)(15,21)(16,19)(18,24)(22,26)(23,25);(11,14,16,23)(12,18,15,22)(13,20,19,25)(17,24,21,26);(3,5,6,9)(4,8,7,10)(11,15,16,12)(13,21,19,17)(14,22,23,18)(20,26,25,24)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55773);