#6686589

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=f,e^2=fi,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, 6593785);