C₂×#19890 #6358257

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3272	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2104 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265453);