#7081588

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	#59093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 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)(17,18,22,27)(19,24,28,32)(20,26,21,25)(23,29,31,30);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,30,22,28,20,29)(18,23,26,24,27,31,25,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,28)(23,31)(24,32)(29,30);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,19)(18,24)(20,30)(21,29)(22,28)(23,26)(25,31)(27,32);(10,14)(13,15)(17,20,22,21)(18,25,27,26)(19,29,28,30)(23,32,31,24)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=hi,d^2=g,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,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, 6988784);