Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C4×C5×AGL(1,ℤ/8) (order 640) · Inn(G) = D4 · Out(G) = C22×C4×C5 |
| homology | Abelianization (H1) = C22×C25 · Schur multiplier (H2) = C2 |
| direct product | D8×C25 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | D4 |
| commutator | C4 | C22×C25 |
| Frattini, cosocle | C4×C5 | C22×C5 |
| socle | C2×C5 | C5×D4 |
| prime radical | C5×D8 | C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ C4×C25 ◁ G | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33);(2,8)(3,7)(4,6)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 25);