#143467

order	512
nilpotency class	4
isoclinism	isoclinic class of #143446
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6)(3,9)(4,13)(8,14)(10,15)(11,12)(17,18)(19,29)(20,24)(21,38)(22,40)(23,28)(25,32)(26,34)(27,36)(30,48)(31,47)(33,46)(35,44)(37,43)(39,45)(41,42);(1,2,5,8)(3,10,12,15)(4,14,13,6)(7,11,16,9)(18,24)(19,30)(20,34)(21,36)(22,37)(26,27)(28,43)(29,46)(31,39)(32,41)(33,35)(38,40)(42,44)(47,48);(1,3)(2,7)(4,11)(5,12)(6,15)(8,16)(9,13)(10,14)(17,19,23,33)(18,25,28,45)(20,35,37,30)(21,39,36,32)(22,41,34,31)(24,42,43,47)(26,44,40,48)(27,46,38,29)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=gi,e^2=gh,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fgi,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, 50663);