#6347349

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2760 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,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, 6254545);