C₂×#443 #2953

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×#72214076 (order 2048) · Inn(G) = D₄ · Out(G) = C₂×#2403
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

236 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1685);