#6347517

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347491
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	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2893 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)(3,11,14,15)(4,9,5,8)(7,13,16,12)(18,23)(20,21)(24,31)(25,26)(27,29)(30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,32)(29,31);(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,31)(26,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,16,15);(1,4,6,5)(2,8,10,9)(3,13,14,12)(7,15,16,11)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30)

Polycyclic presentation:

2^9|a^2=hi,d^2=hi,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,e^b=eh,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, 6254713);