Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = C2×B(4,𝔽2) (order 128) · Inn(G) = #99 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| affine special semilinear | AΣL(1,𝔽16) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #99 |
| commutator | C23 | C2×C4 |
| Frattini, cosocle | C2×D4 | C22 |
| prime radical | #120 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,2)(3,7,8,5)(4,6);(2,5)(6,7)Polycyclic presentation:
2^6|a^2=d,b^a=bc,c^a=ce,e^a=ef,d^b=de,d^c=df13 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AΣL(1,𝔽16), #2120, #2121, … |
| maximal normal subgroup of | C2×AΣL(1,𝔽16), #2120, #2121, … |
| maximal quotient of | #1316, #1326, #1343, … |
| outer automorphism group of | #562, #1829, #1830 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 32);