#6351872

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351869
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	#62834	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2977 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)(3,11,12,16)(4,10,5,9)(7,14,15,13)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3)(2,7)(4,13)(5,14)(6,12)(8,15)(9,11)(10,16)(20,29)(24,28)(25,32)(27,30);(4,5)(9,10)(11,16)(13,14)(17,19)(18,22)(20,30)(21,26)(23,31)(24,32)(25,28)(27,29);(2,8)(3,12)(9,10)(13,14)(18,23)(22,31)(24,28)(25,32);(1,4,6,5)(2,9,8,10)(3,13,12,14)(7,11,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259068);