#143335

order	512
nilpotency class	4
isoclinism	isoclinic class of #143209
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	#532	D₄
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

434 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=eh,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, 50531);