#6589727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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

1564 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)(17,18,22,27)(19,24,31,41)(20,32,34,25)(21,36,38,26)(23,28,30,45)(29,43,47,40)(33,48,37,42)(35,46,39,44);(2,6)(7,8)(17,19,22,31)(18,24,27,41)(20,33,34,37)(21,23,38,30)(25,42,32,48)(26,45,36,28)(29,39,47,35)(40,46,43,44);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,34)(18,25,27,32)(19,29,31,47)(21,37,38,33)(23,39,30,35)(24,40,41,43)(26,42,36,48)(28,46,45,44);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,38)(18,26,27,36)(19,30,31,23)(20,29,34,47)(24,28,41,45)(25,43,32,40)(33,35,37,39)(42,46,48,44);(1,4,5,3)(2,8,6,7)(19,21)(23,30)(24,36)(26,41)(28,45)(29,37)(31,38)(33,47)(35,39)(40,48)(42,43)(44,46)

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6496923);