#6498297

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,33)(28,37)(29,31)(30,35)(32,38)(34,39)(36,40);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,28,35,29,36,38,39)(26,30,32,33,31,34,37,40);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,26)(27,34)(28,32)(29,31)(30,36)(33,39)(35,40)(37,38);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,26)(27,30)(28,32)(29,31)(33,35)(34,36)(37,38)(39,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,31)(30,34)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

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, 6405493);