#151567

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	#33853	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

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

normal subgroups only

This group has 241 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,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, 58763);