C₂²×Q₁₆ #571

order	64
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = #72258595 (order 12288) · Inn(G) = D₄ · Out(G) = #72209653
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×Q₁₆

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

129 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2);(3,4);(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14);(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 252);