Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C4×C5×S4 (order 960) · Inn(G) = C22 · Out(G) = C2×C4×C5×D3 |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C22 |
| direct product | #41×C25 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C25 | C22 |
| commutator | C2 | C23×C25 |
| Frattini, socle, cosocle | C2×C5 | C23×C5 |
| prime radical | C5×#41 | C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C25 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,19,11,13,16,20,24,14,17,21,25,28,18,22,26,29,31,23,27,30,32,33);(2,5)(6,8);(1,3,4,7)(2,6,5,8)250 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 48);