#6349552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349529
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	C₂×#3511	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2919 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,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,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, 6256748);