#143285

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	C₂³	#449
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

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

normal subgroups only

This group has 370 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:

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

Polycyclic presentation:

2^9|a^2=i,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,g^a=gi,c^b=cf,d^b=dg,e^b=eh,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, 50481);