#6351657

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351632
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2628 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:

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

Polycyclic presentation:

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

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, 6258853);