C₂²×SD₁₆ #570

order	64
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = #72233730 (order 6144) · Inn(G) = D₄ · Out(G) = #11715850
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×SD₁₆

	subgroup	G/H
center, socle	C₂³	D₄
commutator, Frattini, cosocle	C₄	C₂⁴
prime radical	C₂²×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

149 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 12 points:

(1,2);(3,4);(5,6,8,11)(7,9,12,10);(6,9)(7,12)(10,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 251);