C₂×#334 #1562

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×#72214923 (order 4096) · Inn(G) = D₄ · Out(G) = C₂×#32898
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

100 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 294);