#6686098

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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	#60935	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 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)(20,33)(21,26)(22,27)(23,29)(25,32)(28,36)(30,39)(31,40)(34,43)(35,42)(37,44)(38,45)(41,46)(47,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,21)(22,34)(23,32)(25,29)(26,33)(27,43)(28,40)(30,37)(31,36)(35,46)(38,47)(39,44)(41,42)(45,48);(17,20,21,32,23,31,36,19)(18,25,28,33,29,24,26,40)(22,34,35,46,38,47,37,30)(27,41,44,43,45,39,42,48);(2,7)(6,9)(8,15)(14,16)(17,21,23,36)(18,26,29,28)(19,20,32,31)(22,35,38,37)(24,33,25,40)(27,42,45,44)(30,34,46,47)(39,43,41,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22)(18,27)(19,30)(20,34)(21,35)(23,38)(24,39)(25,41)(26,42)(28,44)(29,45)(31,47)(32,46)(33,43)(36,37)(40,48)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=i,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,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593294);