#137603

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

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	#3035	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

436 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)(3,6)(7,8)(9,10,15,19,13,21,31,24,16,20,34,39,14,18,23,38)(11,22,17,35,25,12,27,29,26,28,36,40,32,30,37,33);(1,2,6,8,4,7,3,5)(9,11,23,37,14,32,34,36,16,26,31,27,13,25,15,17)(10,12,24,40,18,22,19,29,20,30,38,35,21,28,39,33);(9,12)(10,17)(11,24)(13,28)(14,22)(15,33)(16,30)(18,37)(19,32)(20,36)(21,27)(23,40)(25,39)(26,38)(29,34)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44799);