#6590641

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

1673 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)(18,23)(24,38)(25,39)(26,36)(27,41)(40,47)(42,48)(43,44);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,18)(19,27)(20,25)(21,23)(22,36)(24,30)(26,35)(28,42)(29,41)(31,40)(32,43)(33,39)(34,44)(37,38)(45,48)(46,47);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,19,21,29)(18,24,23,38)(20,31,33,46)(22,37,35,30)(25,42,39,48)(26,41,36,27)(28,34,45,32)(40,43,47,44);(9,12,13,11)(10,16,15,14)(17,20)(18,25)(19,28)(21,33)(22,32)(23,39)(24,40)(26,44)(27,42)(29,45)(30,31)(34,35)(36,43)(37,46)(38,47)(41,48);(1,4,5,3)(2,6,8,7)(18,26)(20,32)(22,35)(23,36)(24,41)(25,39)(27,38)(28,31)(30,37)(33,34)(42,48)(45,46)

Polycyclic presentation:

2^9|b^2=fgi,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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, 6497837);