#145915

order	512
nilpotency class	4
isoclinism	isoclinic class of #145904
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	#464
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 223 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53111);