#6555919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#29802	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6463115);