C₂×#529 #3439

order	128
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center	C₂²×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵
socle	C₂³	C₂⁴
prime radical	C₂×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

616 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)(13,15)(14,16)(17,18);(4,7)(8,10)(11,13,14,17)(12,15,16,18);(12,16)(15,18);(3,5,6,9)(4,8,7,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2171);