#6682158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	#36101	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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)(3,5)(4,6)(7,8)(17,18,22,25)(19,26,28,23)(20,29,21,24)(27,31,30,32);(1,3,4,7)(2,5,6,8)(9,10,13,16)(11,15,12,14)(17,19,22,28)(18,23,25,26)(20,27,21,30)(24,31,29,32);(2,6)(5,8)(10,14)(11,12)(15,16);(9,11,13,12)(10,15,16,14)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(10,15)(11,12)(14,16)(17,21,22,20)(18,24,25,29)(19,27,28,30)(23,32,26,31)

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589354);