C₂×#544 #3457

order	128
nilpotency class	2
isoclinism	isoclinic class of #534
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	C₂⁴	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

540 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)(5,9)(6,8)(7,10)(11,12,14,16)(13,17,18,15);(3,4,6,8)(5,7,10,9);(3,4)(5,7)(6,8)(9,10)(11,13,14,18)(12,15,16,17);(3,5)(4,7)(6,10)(8,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2189);