#141716

order	512
nilpotency class	4
isoclinism	isoclinic class of #141715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×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	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 320 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eh,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, 48912);