#6544589

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26,31,34)(27,33,38,40)(28,32,37,39)(29,35,30,36);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,27,31,38)(26,32,34,39)(28,29,37,30)(33,36,40,35);(11,18)(14,17)(19,24)(21,23)(25,28)(26,33)(27,29)(30,38)(31,37)(32,36)(34,40)(35,39);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(19,24)(21,23)(26,34)(32,39)(33,40)(35,36);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,31,30)(26,35,34,36)(27,37,38,28)(32,33,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451785);