#6742722

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#25575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,22)(19,24,30,29)(20,25,33,37)(21,26,36,35)(27,38,43,31)(28,39,46,45)(32,40,34,41)(42,48,44,47);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19,23,30)(18,24,22,29)(20,31,33,38)(21,28,36,46)(25,27,37,43)(26,39,35,45)(32,47,34,48)(40,42,41,44);(10,14)(13,15)(17,20,23,33)(18,25,22,37)(19,27,30,43)(21,34,36,32)(24,38,29,31)(26,41,35,40)(28,44,46,42)(39,47,45,48);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21,23,36)(18,26,22,35)(19,28,30,46)(20,32,33,34)(24,39,29,45)(25,40,37,41)(27,42,43,44)(31,47,38,48);(1,5,6,4)(2,7,3,8)(11,16)(13,15)(18,22)(19,29)(20,33)(24,30)(26,35)(27,38)(28,45)(31,43)(32,34)(39,46)(42,48)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6649918);