#6621150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528346);