#136591

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
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×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

423 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:

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

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^b=dg,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43787);