#6586809

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,30,40)(20,24,34,44)(21,25,36,46)(27,37,47,48)(28,38,29,39)(31,41,32,42)(33,43,35,45);(2,5)(6,8)(17,19,22,30)(18,23,26,40)(20,31,34,32)(21,28,36,29)(24,41,44,42)(25,38,46,39)(27,35,47,33)(37,45,48,43);(10,13)(14,16)(17,20,22,34)(18,24,26,44)(19,27,30,47)(21,35,36,33)(23,37,40,48)(25,45,46,43)(28,31,29,32)(38,41,39,42);(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,32)(21,28)(22,30)(24,42)(25,38)(26,40)(27,35)(29,36)(31,34)(33,47)(37,45)(39,46)(41,44)(43,48);(1,3,4,7)(2,6,5,8)(19,28)(21,36)(23,38)(25,46)(27,31)(29,30)(32,47)(33,35)(37,41)(39,40)(42,48)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6494005);