D₄×#364 #497824

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

736 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 20 points:

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

maximal subgroup of	D₄×#1369, D₄×#1336, D₄×#2171, …
maximal normal subgroup of	D₄×#1369, D₄×#1336, D₄×#2171

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405020);