#6556060

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,d^2=fgh,e^2=h,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, 6463256);