#6578287

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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

1994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485483);