#6571511

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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1410 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fgi,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,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, 6478707);