Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C52 · Schur multiplier (H2) = C23×C5 |
| direct product | C52×#458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C52 | #120 |
| commutator, Frattini, cosocle | C23 | C23×C52 |
| prime radical | C52×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52 ◁ C23×C52 ◁ G | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3,4,5)(11,12,15,18)(13,16,20,23)(14,17,21,24)(19,25,26,22);(6,7,8,9,10)(11,13)(12,16)(14,19)(15,20)(17,22)(18,23)(21,26)(24,25);(11,14)(12,17)(13,19)(15,21)(16,22)(18,24)(20,26)(23,25)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5547);