C₄×#108 #1777

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = #72262791 (order 4096) · Inn(G) = D₄ · Out(G) = #4928938
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

114 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,3,4);(5,6,9,12,10,13,8,18)(7,14,16,19,17,20,15,11);(5,7,8,15,10,17,9,16)(6,11,12,14,13,19,18,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 509);