#347227

order	512
nilpotency class	3
isoclinism	isoclinic class of #347226
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2523 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:

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

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,e^d=eh,g^d=gh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254423);