#7081587

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1585 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)(17,18,22,20)(19,25,27,32)(21,30,26,24)(23,31,29,28);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,26,31,22,27,21,28)(18,23,30,32,20,29,24,25);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(18,20)(19,27)(24,30)(28,31);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,22,18)(19,25,27,32)(21,24,26,30)(23,31,29,28);(10,14)(13,15)(18,20)(23,29)(24,30)(25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=g,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,e^d=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6988783);