C₂×#2031 #23471

order	256
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center	C₂×C₄	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	#393
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

617 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)(5,12,14,18)(6,10)(7,16)(8,11)(9,15,17,13);(5,13,14,15)(9,12,17,18);(3,5)(4,9)(6,13)(7,15)(8,14)(10,18)(11,17)(12,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16459);