#6710261

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1304 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,31)(27,34,37,32)(28,33,29,39)(35,38,40,36);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,37)(26,32,31,34)(28,38,29,36)(33,35,39,40);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,30,29)(26,33,31,39)(27,35,37,40)(32,36,34,38);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,30,28)(26,33,31,39)(27,36,37,38)(32,35,34,40);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,31)(27,32)(33,39)(34,37)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,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, 6617457);