#6347330

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2784 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)(3,10)(4,9)(5,14)(6,8)(7,12)(11,15)(13,16)(17,18,22,25)(19,26,29,31)(20,24,21,30)(23,28,32,27);(1,3,6,13)(2,7,8,15)(4,12,5,11)(9,16,14,10)(17,19,22,29)(18,23,25,32)(20,27,21,28)(24,31,30,26);(17,18)(19,23)(20,24)(21,30)(22,25)(26,28)(27,31)(29,32);(2,8)(7,15)(9,14)(10,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,32,26);(1,4,6,5)(2,9,8,14)(3,11,13,12)(7,10,15,16)(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,26,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=i,e^2=h,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, 6254526);