C₄×#398 #13863

order	256
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

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

787 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)(15,20)(16,18)(17,19);(9,10)(11,12)(13,15,17,16)(14,18,19,20);(6,8)(10,12)(13,16,17,15)(14,18,19,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6851);