#7090163

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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	#23863	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1746 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,e^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=eh,d^c=dh,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, 6997359);