#6591333

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1331 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,c^2=fg,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498529);