#143341

order	512
nilpotency class	4
isoclinism	isoclinic class of #143216
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28495	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

440 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ehi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50537);