#198971

order	512
nilpotency class	3
isoclinism	isoclinic class of #198807
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

1239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106167);