#145434

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	#2034
prime radical	C₂³	#465
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

268 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,10)(3,11,13,16)(4,8,12,7)(5,15,14,9)(17,18)(19,27)(20,31)(21,36)(22,25)(23,29)(24,34)(26,33)(28,43)(30,40)(32,39)(35,47)(37,44)(38,45)(41,48)(42,46);(1,3,6,13)(2,7,10,8)(4,14,12,5)(9,16,15,11)(23,38)(27,43)(39,42)(40,44);(1,4,5,3,6,12,14,13)(2,8,9,16,10,7,15,11)(17,19,21,37,22,28,33,30)(18,23,26,42,25,38,36,39)(20,32,34,45,35,46,48,29)(24,40,31,43,41,44,47,27)

Polycyclic presentation:

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

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, 52630);