#6346740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346727
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	#62819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2954 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6253936);