#6587604

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,45)(36,44)(37,46)(47,48);(1,3,5,4)(2,6,7,8)(17,19,23,32)(18,24,27,41)(20,33,35,47)(21,30,31,22)(25,42,45,48)(26,39,40,28)(29,37,34,36)(38,46,43,44);(10,13)(14,16)(17,20,23,35)(18,25,27,45)(19,29,32,34)(21,33,31,47)(22,37,30,36)(24,38,41,43)(26,42,40,48)(28,46,39,44);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,34)(22,32)(23,31)(24,39)(25,43)(27,40)(28,41)(29,35)(33,36)(37,47)(38,45)(42,44)(46,48);(1,4,5,3)(2,6,7,8)(18,27)(19,21)(20,35)(22,30)(24,40)(26,41)(29,47)(31,32)(33,34)(38,42)(43,48)(44,46)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=ehi,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, 6494800);