#6555473

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

Polycyclic presentation:

2^9|a^2=fi,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, 6462669);