#6359045

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359034
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61091	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh

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, 6266241);