#6349683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349612
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2648 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,16)(14,15)(17,18)(19,28)(20,29)(21,27)(22,26)(23,25)(24,32)(30,31);(1,3,6,14)(2,7,10,16)(4,12,5,13)(8,15,9,11)(17,19,23,24)(18,20,25,31)(21,30,22,29)(26,32,27,28);(1,2)(3,7)(4,9)(5,8)(6,10)(11,12)(13,15)(14,16)(17,20)(18,24)(19,25)(21,32)(22,28)(23,31)(26,29)(27,30);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(18,25)(20,31)(26,27)(28,32);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(18,25)(19,24)(26,27)(29,30)

Polycyclic presentation:

2^9|b^2=h,d^2=hi,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,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, 6256879);