#6499257

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406453);