#135901

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	#1850
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dg,e^b=eg,d^c=dg,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, 43097);