#6684975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6592171);