order	64
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) = #72213937 (order 1024) · Inn(G) = D₈ · Out(G) = #525
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₁₆

Subgroups

	subgroup	G/H
center, socle	C₂²	D₈
commutator, Frattini, cosocle	C₈	C₂³
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

49 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2);(3,4)(5,8)(6,15)(7,11)(9,12)(10,13)(14,17)(16,18);(4,8)(5,12)(6,13)(9,10)(11,17)(14,18)(15,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 186);