#6448079

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3045	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355275);