#6555457

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(17,18,21,26,22,25,36,40)(19,27,30,38,29,41,34,23)(20,31,32,39,33,42,35,24)(28,37,45,48,44,47,46,43);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,25,38)(20,28,33,44)(21,34,36,30)(24,37,39,47)(26,41,40,27)(31,43,42,48)(32,46,35,45);(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);(9,11)(10,14)(12,15)(13,16)(19,29)(23,38)(27,41)(28,44)(30,34)(37,47)(43,48)(45,46);(11,15)(14,16)(18,25)(23,38)(24,39)(26,40)(27,41)(31,42)(37,47)(43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,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, 6462653);