#6547701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1243 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:

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

Polycyclic presentation:

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

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, 6454897);