#6690186

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,28)(20,25)(21,31)(22,32)(23,27)(24,29)(26,30);(9,12,13,11)(10,14,15,16)(17,19,23,26)(18,24,27,21)(20,29,22,31)(25,30,32,28);(2,6)(5,7)(17,20,23,22)(18,25,27,32)(19,29,26,31)(21,28,24,30);(3,8)(5,7)(17,21,23,24)(18,26,27,19)(20,30,22,28)(25,29,32,31)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6597382);