#6611307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574031
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1516 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)(9,10)(11,14)(12,15)(13,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(9,11)(10,13)(12,16)(14,15)(17,19,20,29,22,27,21,28)(18,23,30,32,25,31,24,26);(1,3,4,8)(2,5,6,7)(11,16)(13,14)(19,27)(23,31)(26,32)(28,29);(9,11)(10,14)(12,16)(13,15)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,32,31,26);(11,16)(13,14)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,32,31,26)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=ehi

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, 6518503);