Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C24×D4 (order 128) · Inn(G) = D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4 |
| commutator | C4 | C2×C8 |
| Frattini, cosocle | C42 | C22 |
| socle, prime radical | C22 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
35 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,5)(3,7)(6,8)(9,10,11,13,12,14,15,16);(1,2,4,6)(3,8,7,5)Polycyclic presentation:
2^6|a^2=d,b^2=e,c^2=e,d^2=f,b^a=bc,c^a=ce,c^b=ce28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×Q8⋊C8, #1476, #1477, … |
| maximal normal subgroup of | C2×Q8⋊C8, #1476, #1477, … |
| maximal quotient of | #1271, #1272, #1279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 7);