C₂×#559 #3479

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

620 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);(3,4,7,6)(5,9,10,8)(12,15)(16,17);(4,6)(8,9)(11,12,14,15)(13,17,18,16);(3,5,7,10)(4,8,6,9);(3,6,7,4)(5,9,10,8)(11,13,14,18)(12,16,15,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2211);