#6349289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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	#62805	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2787 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)(17,18,23,26)(19,27,24,32)(20,28,30,29)(21,25,22,31);(1,3,6,13)(2,7,10,15)(4,12,5,14)(8,11,9,16)(17,19,23,24)(18,20,26,30)(21,29,22,28)(25,32,31,27);(17,20)(18,24)(19,26)(21,27)(22,32)(23,30)(25,29)(28,31);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16);(3,13)(7,15)(11,16)(12,14)(17,21,23,22)(18,25,26,31)(19,28,24,29)(20,27,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256485);