#6359119

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359112
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

1954 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)(3,5)(4,7)(6,8)(9,10,16,21)(11,22,26,38)(12,13,18,31)(14,20,15,19)(17,33,37,25)(23,24,27,40)(28,34,32,30)(29,39,35,36);(9,11,16,26)(10,17,21,37)(12,27,18,23)(13,29,31,35)(14,25,15,33)(19,22,20,38)(24,34,40,30)(28,39,32,36);(2,5)(3,8)(6,7)(9,12,16,18)(10,13,21,31)(11,23,26,27)(14,32,15,28)(17,35,37,29)(19,34,20,30)(22,24,38,40)(25,39,33,36);(9,13)(10,18)(11,24)(12,21)(14,30)(15,34)(16,31)(17,36)(19,32)(20,28)(22,27)(23,38)(25,35)(26,40)(29,33)(37,39);(9,14,16,15)(10,19,21,20)(11,25,26,33)(12,28,18,32)(13,30,31,34)(17,22,37,38)(23,36,27,39)(24,35,40,29)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266315);