#6589658

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#36410	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1735 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6496854);