#6349526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349437
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	#62815	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2823 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)(3,11,14,15)(4,8,5,9)(7,13,16,12)(17,18,23,27)(19,29,25,32)(20,31,24,30)(21,28,22,26);(1,3,6,14)(2,7,10,16)(4,12,5,13)(8,15,9,11)(17,19,23,25)(18,24,27,20)(21,30,22,31)(26,32,28,29);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,23,24)(18,25,27,19)(21,29,22,32)(26,31,28,30);(17,21,23,22)(18,26,27,28)(19,30,25,31)(20,29,24,32);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(18,27)(19,25)(26,28)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256722);