#6681712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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)(11,13)(12,14)(15,16);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,18)(19,28)(20,24)(21,25)(22,27)(23,30)(26,36)(29,44)(31,43)(32,46)(33,40)(34,42)(35,38)(37,45)(39,47)(41,48);(10,14)(13,16)(17,19,22,31)(18,23,27,38)(20,32,34,29)(21,35,36,30)(24,39,42,37)(25,28,26,43)(33,45,48,47)(40,46,41,44);(1,4,6,5)(2,7,3,8)(17,20,22,34)(18,24,27,42)(19,29,31,32)(21,33,36,48)(23,37,38,39)(25,40,26,41)(28,44,43,46)(30,45,35,47);(1,5,6,4)(2,7,3,8)(18,25)(21,36)(23,28)(24,40)(26,27)(30,35)(33,48)(37,44)(38,43)(39,46)(41,42)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6588908);