#6621178

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1414 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)(17,18,21,26,23,28,22,27)(19,29,33,39,31,40,32,24)(20,34,36,25,37,41,35,42)(30,44,46,43,45,38,47,48);(1,3,4,8)(2,5,6,7)(10,14)(12,15)(17,19,23,31)(18,24,28,39)(20,30,37,45)(21,32,22,33)(25,38,42,44)(26,40,27,29)(34,48,41,43)(35,46,36,47);(9,10)(11,12)(13,14)(15,16)(17,20,23,37)(18,25,28,42)(19,30,31,45)(21,35,22,36)(24,38,39,44)(26,34,27,41)(29,43,40,48)(32,46,33,47);(9,11,13,16)(10,15,14,12)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(9,12,13,15)(10,11,14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,34,42,41)(30,46,45,47)(38,48,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528374);