#143345

order	512
nilpotency class	4
isoclinism	isoclinic class of #143219
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

439 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 86 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)(10,16)(11,21)(12,26)(13,14)(17,33)(18,27)(19,31)(20,32)(22,37)(23,39)(24,25)(28,29)(30,34)(35,38);(2,6)(3,4)(7,8)(9,10,15,20)(11,22,25,34)(12,27,29,31)(13,19,14,18)(16,28,32,26)(17,21,35,24)(23,30,39,37)(33,36,38,40);(2,7)(3,4)(6,8)(9,11)(10,17)(12,23)(13,24)(14,21)(15,25)(16,30)(18,22)(19,34)(20,35)(26,36)(27,33)(28,40)(29,39)(31,38)(32,37)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50541);