#201164

order	512
nilpotency class	3
isoclinism	isoclinic class of #201103
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9,6,10,4,8)(3,7,11,14,12,16,13,15)(17,18,22,26,23,27,21,37)(19,28,32,39,31,43,30,24)(20,33,35,41,36,25,34,40)(29,42,46,38,45,48,44,47);(1,3,4,11,6,12,5,13)(2,7,9,15,10,16,8,14)(17,19,23,31)(18,24)(20,29,36,45)(21,30,22,32)(25,38)(26,28)(27,39)(33,47)(34,44,35,46)(37,43)(40,48)(41,42);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,16,15)(17,20,23,36)(18,25,27,33)(19,29,31,45)(21,35,22,34)(24,38,39,47)(26,41,37,40)(28,42,43,48)(30,46,32,44);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,15,16,14)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi

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, 108360);