#6589284

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fg,e^2=f,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, 6496480);