C₄⋊C₄×#123 #519298

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3966 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426494);