#6586145

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,30,39)(20,24,34,43)(21,25,35,44)(27,36,29,38)(28,37,31,40)(32,41,33,42)(45,47,46,48);(1,3,5,4)(2,6,7,8)(17,19,22,30)(18,23,26,39)(20,31,34,28)(21,27,35,29)(24,40,43,37)(25,36,44,38)(32,46,33,45)(41,48,42,47);(9,11,12,16)(10,13,14,15)(19,27)(20,32)(21,35)(23,36)(24,41)(25,44)(29,30)(33,34)(38,39)(42,43)(45,46)(47,48);(1,4,5,3)(2,6,7,8)(17,20)(18,24)(19,28)(21,32)(22,34)(23,37)(25,41)(26,43)(27,45)(29,46)(30,31)(33,35)(36,47)(38,48)(39,40)(42,44);(19,29)(20,32)(21,35)(23,38)(24,41)(25,44)(27,30)(28,31)(33,34)(36,39)(37,40)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493341);