#6684985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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
prime radical	#61088	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1899 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,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,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592181);