#6621166

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1428 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=i,d^2=h,e^2=hi,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, 6528362);