#133867

order	512
nilpotency class	4
isoclinism	isoclinic class of #133847
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

911 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 911 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7,8,5)(4,6)(9,10)(11,20,23,32)(12,17)(13,18,27,16)(14,19)(15,21,30,24)(22,36)(25,33,40,31)(26,34)(28,37)(29,39)(35,38);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,24)(13,22)(14,23)(16,28)(17,32)(18,29)(19,30)(20,34)(21,26)(25,39)(27,38)(31,36)(33,35)(37,40);(10,16)(13,27)(15,28)(17,31)(18,19)(20,35)(22,38)(25,40)(29,30)(32,36)(33,34)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fgi,e^c=ei,f^c=fhi,f^e=fi

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, 41063);