#136222

order	512
nilpotency class	4
isoclinism	isoclinic class of #136222
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43418);