#141732

order	512
nilpotency class	4
isoclinism	isoclinic class of #141715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
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

394 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 102 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)(3,7)(4,8)(5,6)(9,10,14,20,15,18,22,21)(11,16,25,30,17,34,26,36)(12,27,31,39,32,24,37,40)(13,28,29,19,33,23,38,35);(1,3,7,2,5,8,4,6)(9,11,22,25,15,17,14,26)(10,16,20,36,18,34,21,30)(12,28,37,19,32,23,31,35)(13,24,38,40,33,27,29,39);(2,6)(3,8)(9,12,29,34,14,31,33,36,15,32,38,16,22,37,13,30)(10,17,28,40,21,25,35,24,18,11,23,39,20,26,19,27)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eghi,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,d^b=di,f^b=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48928);