#6346755

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	#62834	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3133 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=hi,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, 6253951);