#6349319

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349298
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	#62830	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2855 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)(3,11,13,15)(4,9,5,8)(7,12,16,14)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,18)(19,23)(20,24)(21,28)(22,25)(26,31)(27,32)(29,30);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30);(3,13)(7,16)(11,15)(12,14)

Polycyclic presentation:

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

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, 6256515);