C₂×#324 #1457

order	128
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	Aut(G) = #72258990 (order 4096) · Inn(G) = C₂²⋊C₄ · Out(G) = C₂²×B(4,𝔽₂)
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

164 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 189);