#144893

order	512
nilpotency class	4
isoclinism	isoclinic class of #144892
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

452 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fg,g^b=gi,h^b=hi,e^c=egi,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52089);