#6555852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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)(17,18,21,26,22,25,36,40)(19,27,30,38,29,41,34,23)(20,31,32,39,33,42,35,24)(28,37,45,48,44,47,46,43);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,34)(22,29)(24,37)(25,38)(26,41)(27,40)(30,36)(31,43)(32,46)(33,44)(35,45)(39,47)(42,48);(10,13)(14,16)(17,20,22,33)(18,24,25,39)(19,28,29,44)(21,35,36,32)(23,37,38,47)(26,42,40,31)(27,43,41,48)(30,46,34,45);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(18,25)(19,29)(24,39)(26,40)(28,44)(30,34)(31,42)(45,46);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(19,29)(23,38)(27,41)(28,44)(30,34)(37,47)(43,48)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6463048);