Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C22×C5 |
| direct product | C52×C22⋊C4 |
| unambiguous semidirect product | C22⋊(C4×C52) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C52 | C22 |
| commutator | C2 | C2×C4×C52 |
| Frattini, cosocle | C22 | C22×C52 |
| prime radical | C23×C52 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ G | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4)(5,6)(7,8)(9,10,11,12,13);(6,8)(14,15,16,17,18)250 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 109);