Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | D4 |
| commutator | C4 | C2×C4×C13 |
| Frattini, cosocle | C2×C4 | C22×C13 |
| prime radical | C2×D4×C13 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C4×C13 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14,15,16,17,18,19,20,21,22,23,24,25);(6,9)(7,12)(10,11)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 52);