#7081562

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#59093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6988758);