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×#3524 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
7226 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)(12,13)(15,23)(17,19)(21,26);(11,12,16,13)(14,19,22,17)(15,18,23,20)(21,25,26,24);(3,4)(5,8)(6,7)(9,10)(11,13,16,12)(14,19,22,17)(15,20,23,18)(21,25,26,24);(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25);(3,5,6,9)(4,8,7,10)(11,15)(12,18)(13,20)(14,21)(16,23)(17,24)(19,25)(22,26)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55780);