#6742083

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	#30413	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fgi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6649279);