#198868

order	512
nilpotency class	3
isoclinism	isoclinic class of #198789
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

982 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106064);