#6586619

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1289 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,25)(21,24)(22,26)(27,44)(28,37)(29,36)(30,38)(31,41)(32,43)(33,39)(34,42)(35,40)(45,48)(46,47);(1,3,5,4)(2,6,8,7)(17,19,21,30)(18,23,24,38)(20,31,33,28)(22,27,35,29)(25,41,39,37)(26,44,40,36)(32,46,34,45)(42,48,43,47);(9,11,12,16)(10,13,14,15)(18,24)(19,27)(20,32)(22,35)(23,36)(25,42)(29,30)(33,34)(37,41)(38,44)(39,43)(45,46);(1,3,5,4)(2,7,8,6)(17,20)(18,25)(19,28)(21,33)(22,32)(23,37)(24,39)(26,43)(27,45)(29,46)(30,31)(34,35)(36,47)(38,41)(40,42)(44,48);(1,4,5,3)(2,6,8,7)(19,29)(20,32)(22,35)(23,36)(25,43)(26,40)(27,30)(28,31)(33,34)(37,41)(38,44)(39,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=gi,d^2=f,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, 6493815);