#6586157

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

1489 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)(19,23,28,38)(20,24,33,43)(21,25,36,46)(27,37,29,39)(30,40,34,44)(31,41,47,48)(32,42,35,45);(1,3,5,4)(2,6,7,8)(17,19,22,28)(18,23,26,38)(20,31,33,47)(21,34,36,30)(24,41,43,48)(25,44,46,40)(27,35,29,32)(37,45,39,42);(9,11,12,16)(10,13,14,15)(17,20,22,33)(18,24,26,43)(19,27,28,29)(21,35,36,32)(23,37,38,39)(25,45,46,42)(30,47,34,31)(40,48,44,41);(1,4,5,3)(2,6,7,8)(19,28)(20,32)(21,36)(23,38)(24,42)(25,46)(27,47)(29,31)(33,35)(37,48)(39,41)(43,45);(17,20,22,33)(18,24,26,43)(19,29,28,27)(21,35,36,32)(23,39,38,37)(25,45,46,42)(30,31,34,47)(40,41,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=gi,d^2=f,e^2=i,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, 6493353);