#6690131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,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, 6597327);