#198205

order	512
nilpotency class	3
isoclinism	isoclinic class of #198126
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	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1049 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)(17,18,22,26,23,28,21,27)(19,29,32,39,33,42,31,24)(20,34,36,25,37,40,35,41)(30,38,45,47,46,48,44,43);(1,3,4,11,6,12,5,13)(2,7,9,15,10,16,8,14)(17,19,23,33)(18,24)(20,30,37,46)(21,31,22,32)(25,38)(26,29)(27,42)(28,39)(34,43)(35,44,36,45)(40,47)(41,48);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,16,15)(17,20,23,37)(18,25,28,41)(19,30,33,46)(21,36,22,35)(24,38,39,48)(26,34,27,40)(29,43,42,47)(31,45,32,44);(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,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105401);