#6349728

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2833 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256924);