#6349547

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	#60561	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2951 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)(4,10)(5,9)(6,8)(7,13)(12,15)(14,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,28)(29,32)(30,31);(1,3,6,14)(2,7,8,15)(4,13,5,12)(9,11,10,16)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(2,8)(7,15)(9,10)(11,16);(1,4,6,5)(2,9,8,10)(3,12,14,13)(7,16,15,11)(17,20,22,21)(18,25,26,23)(19,28,30,29)(24,27,32,31)

Polycyclic presentation:

2^9|b^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,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, 6256743);