C₂×#3079 #60482

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	C₂²×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1179 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53470);