Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of C24⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C24⋊C25 |
| derived, Sylow | C24 ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C5 | C24⋊C5 |
| commutator, hypocenter | C24 | C25 |
| Fitting, socle, prime radical, cosocle | C24×C5 | C5 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C5 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 50 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,6,8)(3,5,7,9,10)(11,12,14,17,21,13,15,18,22,26,16,19,23,27,30,20,24,28,31,33,25,29,32,34,35);(1,3)(8,10)Polycyclic presentation:
5^2,2^4|a^5=b,c^a=f,d^a=cf,e^a=cdf,f^a=cdef40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 52);