#145581

order	512
nilpotency class	4
isoclinism	isoclinic class of #143938
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2054
prime radical	C₂³	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 262 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fh,g^b=gi,h^b=hi,e^c=egh,f^c=fgh

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, 52777);