C₂×#2081 #23650

order	256
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂²	#397
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

503 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);(3,4,7,9,8,12,6,11)(5,13,15,18,14,17,16,10);(4,9)(5,14)(6,7)(10,17)(11,12)(13,18);(3,5,6,15,8,14,7,16)(4,10,11,13,12,18,9,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16638);