D₄×#126 #36106

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1254 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29094);