Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#95 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C22 |
| commutator | C2 | C42×C13 |
| Frattini, cosocle | C23 | C22×C13 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4)(9,10)(11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25);(5,6,7,8)(10,12)260 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 45);