C₂×#2068 #23608

order	256
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

603 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16596);