#6524715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499992
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	#62836	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431911);