#138709

order	512
nilpotency class	4
isoclinism	isoclinic class of #138689
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=i,d^2=ghi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45905);