#136621

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

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

394 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,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, 43817);