#6529322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500010
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	C₂⁴	C₂⁵
prime radical	#62823	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2938 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi

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, 6436518);