C₂×#3110 #60542

order	256
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1275 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53530);