C₄×#387 #13765

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×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

871 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)(9,10,12,11)(13,14)(15,18)(16,17)(19,20);(9,11,12,10)(14,17)(18,20);(6,8)(13,15,16,19)(14,18,17,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6753);