#6574177

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2628 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6481373);