#6688645

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1416 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6595841);