#145989

order	512
nilpotency class	4
isoclinism	isoclinic class of #145988
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2067
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#478
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53185);