#6349718

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 6256914);