#141094

order	512
nilpotency class	4
isoclinism	isoclinic class of #141077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 470 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)(9,10,14,20,33,19,32,18,15,21,31,17,13,22,12,30)(11,23,27,16,26,36,25,40,28,38,24,37,29,39,34,35);(1,3,7,2,5,8,4,6)(9,11,13,26,15,28,33,29)(10,16,19,35,21,37,22,40)(12,27,31,25,32,24,14,34)(17,38,30,36,20,23,18,39);(2,6)(3,8)(9,12,13,31,15,32,33,14)(10,17,19,30,21,20,22,18)(11,24,26,34,28,27,29,25)(16,23,35,39,37,38,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48290);