#151538

order	512
nilpotency class	5
isoclinism	isoclinic class of #151520
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
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58734);