#6590422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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

1669 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,20)(19,24,31,32)(21,25,38,33)(22,26,39,34)(27,29,41,30)(28,40,45,47)(35,37,42,36)(43,44,48,46);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,23,31)(18,24,20,32)(21,28,38,45)(22,29,39,30)(25,40,33,47)(26,41,34,27)(35,48,42,43)(36,44,37,46);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20,23,18)(19,27,31,41)(21,35,38,42)(22,26,39,34)(24,29,32,30)(25,37,33,36)(28,40,45,47)(43,46,48,44);(9,12,13,11)(10,16,15,14)(17,21)(18,25)(19,28)(20,33)(22,36)(23,38)(24,40)(26,35)(27,43)(29,44)(30,46)(31,45)(32,47)(34,42)(37,39)(41,48);(1,4,5,3)(2,6,8,7)(19,29)(21,36)(22,39)(24,41)(25,35)(26,34)(27,32)(30,31)(33,42)(37,38)(43,48)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497618);