#6349312

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	#60790	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2689 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)(3,10)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(18,23)(19,26)(20,21)(25,29)(27,28)(30,31);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,18,22,25)(19,24,28,32)(20,23,21,29)(26,30,27,31);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,19,22,28)(18,24,25,32)(20,26,21,27)(23,30,29,31);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(17,20,22,21)(18,23,25,29)(19,27,28,26)(24,31,32,30);(3,13)(8,11)(10,16)(12,15)(18,25)(23,29)(24,32)(30,31)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,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, 6256508);