#6583004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,34)(22,27)(23,28)(26,36)(29,39)(30,41)(31,42)(32,44)(33,45)(35,48)(37,46)(38,40)(43,47);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,23,30)(18,24,28,41)(20,31,33,35)(21,22,36,38)(25,42,45,48)(26,40,34,27)(29,37,47,32)(39,46,43,44);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20,23,33)(18,25,28,45)(19,29,30,47)(21,35,36,31)(22,37,38,32)(24,39,41,43)(26,42,34,48)(27,46,40,44);(1,3,5,4)(2,7,8,6)(17,21)(18,26)(19,22)(20,29)(23,36)(24,40)(25,43)(27,41)(28,34)(30,38)(31,37)(32,35)(33,47)(39,45)(42,44)(46,48);(1,4,5,3)(2,6,8,7)(19,21)(22,38)(24,34)(26,41)(27,40)(29,35)(30,36)(31,47)(32,37)(39,48)(42,43)(44,46)

Polycyclic presentation:

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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490200);