#6607291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

1380 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh

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, 6514487);