#136758

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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	#3106	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

337 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=egi,c^b=ci,d^b=dg,f^b=fg,e^c=egi

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, 43954);