#6537950

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6445146);