#140730

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1970
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=di,f^b=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, 47926);