#143289

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	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 369 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,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, 50485);