#6502490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1543 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:

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

Polycyclic presentation:

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

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, 6409686);