#136223

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 393 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)(9,10,14,21,16,23,15,22)(11,24,26,37,28,39,27,38)(12,18,30,20,13,32,31,19)(17,29,34,25,36,33,35,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17,23,36)(12,29)(13,33)(14,27)(15,26)(16,28)(18,24,32,39)(19,37,20,38)(21,35,22,34)(25,31)(30,40);(1,3)(2,5)(4,7)(6,8)(10,18)(12,30)(13,31)(14,15)(17,24)(19,21)(20,22)(23,32)(25,33)(26,27)(29,40)(34,38)(35,37)(36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=i,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, 43419);