#145388

order	512
nilpotency class	4
isoclinism	isoclinic class of #145365
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
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 ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,5)(4,6)(7,8)(9,10)(11,22)(12,13)(14,18)(15,28)(16,21)(17,24)(19,31)(20,30)(23,39)(25,38)(26,34)(27,37)(29,32)(33,36)(35,40);(2,5)(6,7)(9,11,16,27)(10,17,21,35)(12,22,19,37)(13,25,31,39)(14,24,32,40)(15,33,30,26)(18,36,29,34)(20,38,28,23);(2,6)(4,8)(5,7)(9,12,14,20,15,29,31,21,16,19,32,28,30,18,13,10)(11,23,25,37,26,35,40,34,27,38,39,22,33,17,24,36)

Polycyclic presentation:

2^9|b^2=i,c^2=dh,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,h^b=hi,e^c=eghi,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, 52584);