order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×#72233877 (order 2048) · Inn(G) = C₂² · Out(G) = C₂×#35680
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

C₈×M₁₆

Subgroups

112 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 181);