D₄×#395 #4306366

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#392
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

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

5822 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);(10,12)(13,14,17,16)(15,19,20,18);(6,8)(10,12)(13,15,17,20)(14,18,16,19);(5,6)(7,8)(9,10)(11,12)(13,16,17,14)(15,18,20,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4213562);