#6589719

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,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, 6496915);