#6660604

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660376
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^c=dfh,g^c=gi

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, 6567800);