Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = #72210469 (order 1024) · Inn(G) = C2×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,16,12,13);(2,5)(6,8)(9,11,13,10,14,12,16,15);(1,3,4,7)(2,6,5,8)(9,12,13,15,14,11,16,10)Polycyclic presentation:
2^6|a^2=f,b^2=e,c^2=d,d^2=f,e^2=f,b^a=bd,c^a=ce,d^a=df,e^a=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | #1635, #1636, #1637, … |
| maximal normal subgroup of | #1635, #1636, #1637, … |
| maximal quotient of | #1834, #1933, #1956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 168);