#263677

order	512
nilpotency class	3
isoclinism	isoclinic class of #263675
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1074 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,f^c=fi,g^c=gi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170873);