#6358849

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

2002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266045);