D₄×#381 #520475

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

7774 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 427671);