#6686126

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60710	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593322);