#6686117

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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	#60937	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1637 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,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,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, 6593313);