C₄×#376 #13746

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂²×C₄⁴

Aliases

	subgroup	G/H
center	C₂²×C₄²	C₂²
commutator	C₂	C₂×C₄³
Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1151 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6734);