#6629708

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2180 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25)(13,19)(14,29)(15,17)(16,24)(18,28)(21,34)(22,36)(23,31)(26,40)(27,32)(30,37)(33,35)(38,39);(1,3,4,7)(2,5,6,8)(9,11,13,23,15,21,14,24)(10,16,29,34,17,31,19,20)(12,22,26,37,28,35,27,38)(18,30,40,36,25,39,32,33);(2,6)(5,8)(10,17)(11,21)(18,25)(19,29)(22,35)(23,24)(32,40)(37,38);(9,12,14,27,15,28,13,26)(10,18,19,32,17,25,29,40)(11,22,24,38,21,35,23,37)(16,30,20,33,31,39,34,36);(9,13,15,14)(10,19,17,29)(11,23,21,24)(12,26,28,27)(16,20,31,34)(18,32,25,40)(22,37,35,38)(30,33,39,36)

Polycyclic presentation:

2^9|b^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ci,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536904);