#198921

order	512
nilpotency class	3
isoclinism	isoclinic class of #198843
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1016 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:

(1,2,6,9,7,10,5,3)(4,11,13,8,14,15,12,16)(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);(1,3,6,10,7,9,5,2)(4,11,13,16,14,15,12,8)(17,19)(18,24,27,39)(20,29)(21,31)(22,32)(23,30)(25,38,33,47)(26,28,37,43)(34,45)(35,46)(36,44)(40,48,41,42);(1,4,7,14)(2,8,10,16)(3,11,9,15)(5,12,6,13)(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);(1,5,7,6)(2,9,10,3)(4,12,14,13)(8,15,16,11)(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=eg,c^2=h,d^2=i,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,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, 106117);