#6447320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354516);