#6571352

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#39131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478548);