#6572265

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533063
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3471 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 5 generators on 16 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=eh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479461);