#6589425

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	#36019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,30,39)(20,24,34,43)(21,25,35,44)(27,36,29,38)(28,37,31,40)(32,41,33,42)(45,47,46,48);(2,6)(7,8)(17,19)(18,23)(20,31)(21,27)(22,30)(24,40)(25,36)(26,39)(28,34)(29,35)(32,46)(33,45)(37,43)(38,44)(41,48)(42,47);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(19,27)(20,32)(21,35)(23,36)(24,41)(25,44)(29,30)(33,34)(38,39)(42,43)(45,46)(47,48);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,22,34)(18,24,26,43)(19,28,30,31)(21,32,35,33)(23,37,39,40)(25,41,44,42)(27,45,29,46)(36,47,38,48);(1,4,5,3)(2,8,6,7)(19,29)(20,32)(21,35)(23,38)(24,41)(25,44)(27,30)(28,31)(33,34)(36,39)(37,40)(42,43)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6496621);