#145571

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	#3147	C₂²
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

289 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^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, 52767);