#232939

order	512
nilpotency class	3
isoclinism	isoclinic class of #232939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1595 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140135);