#136579

order	512
nilpotency class	3
isoclinism	isoclinic class of #9159
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43775);