#6686586

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18)(19,24)(20,32)(21,36)(22,26)(23,27)(25,35)(28,42)(29,37)(30,40)(31,41)(33,47)(34,44)(38,43)(39,46)(45,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,42)(26,29)(27,41)(28,35)(32,47)(34,46)(36,40)(38,45)(39,44)(43,48);(17,20,21,35,23,34,22,38)(18,25,26,32,27,43,36,44)(19,28,30,46,31,45,37,33)(24,39,29,42,41,47,40,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(20,34)(25,43)(28,45)(32,44)(33,46)(35,38)(39,47)(42,48);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,18)(19,29)(20,25)(21,26)(22,36)(23,27)(24,37)(28,42)(30,41)(31,40)(32,35)(33,39)(34,43)(38,44)(45,48)(46,47)

Polycyclic presentation:

2^9|c^2=g,d^2=f,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, 6593782);