C₂×#1712 #19601

order	256
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

	subgroup	G/H
center, socle	C₂³	#127
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

671 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12589);