C₄×#384 #13772

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

	subgroup	G/H
center	C₂³×C₄	C₂³
commutator	C₂²	C₂²×C₄²
Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1047 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6760);