#136270

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

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

normal subgroups only

This group has 390 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=gh,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, 43466);