#6572214

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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)(25,26,31,27)(28,36,38,40)(29,33,30,34)(32,39,35,37);(2,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,27,31,26)(28,35,38,32)(29,33,30,34)(36,37,40,39);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28,31,38)(26,32,27,35)(29,37,30,39)(33,40,34,36);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,29,31,30)(26,33,27,34)(28,37,38,39)(32,40,35,36);(1,4,5,3)(2,8,6,7)(12,20)(15,22)(18,24)(21,23)(25,29,31,30)(26,34,27,33)(28,37,38,39)(32,36,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=fi,e^2=fi,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, 6479410);