#6567952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6475148);