C₂×#2055 #23537

order	256
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

851 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16525);