D₄×#446 #6385076

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

2124 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

maximal subgroup of	D₄×#3373, D₄×#3399, D₄×#3384, …
maximal normal subgroup of	D₄×#3373, D₄×#3399, D₄×#3384, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6292272);