#151544

order	512
nilpotency class	5
isoclinism	isoclinic class of #151512
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#12430	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

202 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,h^a=hi,d^b=df,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58740);