#6570276

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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

1345 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,12,13,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,41)(29,38)(30,39)(31,45)(33,40)(35,46)(37,44)(42,48)(43,47);(1,3,5,4)(2,6,8,7)(9,11,15,13)(10,16,14,12)(17,19,23,29)(18,24,27,38)(20,28,35,42)(21,30,22,31)(25,37,40,47)(26,39,36,45)(32,41,46,48)(33,43,34,44);(9,12,15,16)(10,11,14,13)(17,20,21,33,23,35,22,34)(18,25,36,46,27,40,26,32)(19,28,30,43,29,42,31,44)(24,37,45,48,38,47,39,41);(1,3,5,4)(2,7,8,6)(9,13,15,11)(10,16,14,12)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44);(1,4,5,3)(2,6,8,7)(9,14,15,10)(11,12,13,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,35,34)(24,39,38,45)(25,32,40,46)(28,43,42,44)(37,41,47,48)

Polycyclic presentation:

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

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, 6477472);