#6528927

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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)(25,26,31,29)(27,35,38,40)(28,34,30,33)(32,37,39,36);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,38)(26,32,29,39)(28,36,30,37)(33,35,34,40);(2,5)(6,8)(11,17)(12,19)(14,20)(15,22)(25,28,31,30)(26,33,29,34)(27,36,38,37)(32,35,39,40);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29,31,26)(27,32,38,39)(28,34,30,33)(35,37,40,36);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,31,30)(26,34,29,33)(27,37,38,36)(32,35,39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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, 6436123);