#198301

order	512
nilpotency class	3
isoclinism	isoclinic class of #198135
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1225 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)(11,21)(12,25,28,33)(13,18)(14,19)(15,20)(16,23)(17,27,34,26)(22,36,38,30)(24,31)(29,32)(35,37,40,39);(2,6)(3,4)(7,8)(9,11,14,29,15,24,13,23)(10,16,18,31,20,32,19,21)(12,22,26,37,28,38,27,39)(17,30,25,40,34,36,33,35);(2,6)(3,4)(7,8)(9,12)(10,17)(11,22)(13,27)(14,26)(15,28)(16,30)(18,25)(19,33)(20,34)(21,35)(23,39)(24,38)(29,37)(31,40)(32,36);(2,7)(6,8)(9,13,15,14)(10,18,20,19)(11,23,24,29)(12,26,28,27)(16,31,32,21)(17,33,34,25)(22,37,38,39)(30,35,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,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, 105497);