#6528922

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

1368 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)(25,26,29,31)(27,32,34,40)(28,36,38,33)(30,37,39,35);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,34)(26,30,31,39)(28,37,38,35)(32,33,40,36);(2,5)(6,8)(11,17)(12,19)(14,20)(15,22)(26,31)(30,39)(32,40)(33,36);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,27)(26,32)(28,35)(29,34)(30,36)(31,40)(33,39)(37,38);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(27,34)(30,39)(32,40)(35,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=h,e^2=gi,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, 6436118);