#6508248

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1759 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh

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, 6415444);