D₄×#399 #4306397

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5666 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4213593);