#6730856

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#30462	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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)(11,15)(12,16)(13,14)(17,18,22,20)(19,26,28,24)(21,30,23,25)(27,29,32,31);(1,3,4,8)(2,5,6,7)(17,19,22,28)(18,24,20,26)(21,29,23,31)(25,32,30,27);(10,14)(15,16)(17,20,22,18)(19,27,28,32)(21,30,23,25)(24,31,26,29);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,20)(24,26)(25,30)(27,32);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(17,21)(18,25)(19,28)(20,30)(22,23)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6638052);