C₂×#356 #2131

order	128
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	Aut(G) = #72238724 (order 2048) · Inn(G) = #99 · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

120 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 863);