#6698384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1240 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(27,35)(31,39)(36,40)(37,38);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26,29,34)(27,36,35,40)(28,32,30,33)(31,38,39,37);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,29,35)(26,31,34,39)(28,37,30,38)(32,40,33,36);(1,3,4,7)(2,6,5,8)(10,16)(14,17)(15,22)(21,23)(25,28,29,30)(26,32,34,33)(27,37,35,38)(31,40,39,36);(11,19)(14,17)(18,24)(21,23)(26,33)(28,30)(31,36)(32,34)(37,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6605580);