#6347122

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347092
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	C₂²×C₄	C₂⁵
prime radical	#62686	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2891 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:

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

Polycyclic presentation:

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

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, 6254318);