#6351526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351503
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
prime radical	C₂×#3559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2808 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)(3,11)(4,8)(5,10)(6,9)(7,12)(13,15)(14,16)(17,18,22,19)(20,27,28,32)(21,26,31,24)(23,30,25,29);(1,3)(2,7)(4,14)(5,12)(6,13)(8,11)(9,16)(10,15)(17,19,22,18)(20,25,28,23)(21,24,31,26)(27,30,32,29);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,27)(29,31);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,16,11);(2,9)(7,16)(8,10)(11,15)

Polycyclic presentation:

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

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, 6258722);