#145828

order	512
nilpotency class	4
isoclinism	isoclinic class of #145798
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2058
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

347 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dgh,e^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fi,d^c=di,e^c=eg,g^c=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53024);