#6556998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1304 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,30,32)(27,33,34,40)(28,29,36,37)(31,38,39,35);(2,6)(7,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,29,35,30,34,37,38)(26,31,28,33,32,39,36,40);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,22)(18,23)(25,28,29,32,30,36,37,26)(27,33,35,39,34,40,38,31);(1,4,5,3)(2,8,6,7)(12,19)(16,22)(18,23)(20,24)(27,34)(31,39)(33,40)(35,38);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)

Polycyclic presentation:

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

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, 6464194);