#144487

order	512
nilpotency class	4
isoclinism	isoclinic class of #142392
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=di,c^2=gh,e^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,e^c=egi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51683);