#144461

order	512
nilpotency class	4
isoclinism	isoclinic class of #142308
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 508 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=di,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,d^c=di,e^c=egi,f^c=fi,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, 51657);