#141790

order	512
nilpotency class	4
isoclinism	isoclinic class of #141653
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ehi,d^2=gi,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48986);