#6583066

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3110 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)(6,8)(9,10,15,20)(11,16,23,32)(12,17,24,33)(13,18,28,37)(14,19,22,31)(21,30,26,35)(25,34,39,40)(27,36,29,38);(1,2,3,4)(11,13)(12,24)(14,22)(16,18)(17,33)(19,31)(21,39)(23,28)(25,26)(30,40)(32,37)(34,35);(5,6,7,8)(9,11)(10,16)(12,25)(13,22)(14,28)(15,23)(17,34)(18,31)(19,37)(20,32)(21,29)(24,39)(26,27)(30,38)(33,40)(35,36);(9,12,15,24)(10,17,20,33)(11,21,23,26)(13,25,28,39)(14,29,22,27)(16,30,32,35)(18,34,37,40)(19,38,31,36);(9,13,15,28)(10,18,20,37)(11,22,23,14)(12,26,24,21)(16,31,32,19)(17,35,33,30)(25,27,39,29)(34,36,40,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490262);