#6528857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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

1391 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,34)(29,35)(30,39)(31,33)(32,37)(38,40);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,28)(26,32,33,40)(29,37,30,38)(34,39,36,35);(2,5)(6,8)(11,17)(12,19)(14,20)(15,22)(26,33)(27,28)(35,39)(37,38);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,28,31,27)(26,34,33,36)(29,37,30,38)(32,39,40,35);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,29,31,30)(26,35,33,39)(27,37,28,38)(32,34,40,36)

Polycyclic presentation:

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

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, 6436053);