#6486683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469989
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6393879);