#6351580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351559
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	C₄²	C₂⁵
prime radical	C₂×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2988 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)(3,11)(4,8)(5,10)(6,9)(7,13)(12,15)(14,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(1,3)(2,7)(4,13)(5,12)(6,14)(8,16)(9,15)(10,11)(20,21)(24,29)(26,31)(27,30);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,11,15,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(2,9)(7,15)(8,10)(11,16)(18,25)(23,32)(24,29)(26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258776);