Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×C5×#2832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
2238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(27,28,29,30,31);(4,7)(8,10)(11,12)(13,15)(14,16)(17,18)(19,20,23,22)(21,24,25,26);(5,9)(8,10)(12,13)(16,17)(20,22)(24,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,16,18,17)(19,21,23,25)(20,24,22,26);(3,5)(4,8)(6,9)(7,10)(11,14,15,18)(12,16,13,17)(19,22,23,20)(21,24,25,26)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29653);