#6347345

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347305
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	#62793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

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

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, 6254541);