#6582918

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,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, 6490114);