#199055

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,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, 106251);