C₄×#557 #34169

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂²	C₂⁴×C₄
Frattini, socle, prime radical, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1110 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27157);