#6590452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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

1678 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);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,18,22,27)(19,28,30,39)(20,24,35,43)(21,25,37,26)(23,36,40,31)(29,45,46,41)(32,48,47,38)(33,42,34,44);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,19,22,30)(18,23,27,40)(20,32,35,47)(21,36,37,31)(24,41,43,45)(25,28,26,39)(29,33,46,34)(38,44,48,42);(9,12,13,11)(10,16,15,14)(17,20)(18,24)(19,29)(21,33)(22,35)(23,38)(25,42)(26,44)(27,43)(28,45)(30,46)(31,47)(32,36)(34,37)(39,41)(40,48);(1,4,5,3)(2,6,8,7)(18,25)(19,30)(20,33)(21,37)(23,39)(26,27)(28,40)(29,32)(34,35)(41,45)(42,44)(46,47)

Polycyclic presentation:

2^9|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,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497648);