C₂×#2048 #23523

order	256
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂²	#394
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

725 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16511);