Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C24 |
| commutator, Frattini, cosocle | C2 | C24×C13 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,7)(4,6)(5,8)(9,10,11,12,13,14,15,16,17,18,19,20,21);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7);(2,6)(5,7)221 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 231);