order	64
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) = #10576928 (order 512) · Inn(G) = D₈ · Out(G) = C₂×#41
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

C₂×SD₃₂

Subgroups

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

45 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);(3,4,7,11)(5,8,14,17)(6,15,13,10)(9,18,16,12);(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, 187);