#6353521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353473
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,7,4)(3,10,13,16)(5,9,6,14)(8,15,11,12)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32);(1,3,7,13)(2,8,4,11)(5,12,6,15)(9,16,14,10)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,2,7,4)(3,8,13,11)(5,9,6,14)(10,12,16,15)(17,19,21,26)(18,22,24,31)(20,29,27,25)(23,32,28,30);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260717);