Finite Group DatabasebetaFinite Group Databasebeta| order | 800 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C52 · Schur multiplier (H2) = C2×C5 |
| direct product | C52×#125 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C52 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C52 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ G | |
224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3,4,5)(13,17)(15,18)(19,20,22,24)(21,23,25,26);(6,7,8,9,10)(11,12,14,16)(13,15,17,18)(19,20)(21,23)(22,24)(25,26);(11,13)(12,15)(14,17)(16,18)(19,21,22,25)(20,23,24,26)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(800, 944);