#6531112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500148
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	C₂²×C₄	C₂⁵
prime radical	#62831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2746 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,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, 6438308);