#346794

order	512
nilpotency class	3
isoclinism	isoclinic class of #346786
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

2466 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)(3,7)(6,8)(10,13)(11,15)(14,16);(1,2)(3,8)(4,6)(5,7)(11,15)(13,16);(9,10,12,14)(11,16,15,13);(3,7)(5,8)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253990);