#6347341

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2836 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=hi,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, 6254537);