#6578039

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

1864 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)(3,7)(4,8)(5,6)(9,10)(11,20)(12,17)(13,18)(14,29)(15,19)(16,24)(21,23)(22,35)(25,32)(26,28)(27,33)(30,34)(31,39)(36,38)(37,40);(2,6)(7,8)(9,10,13,18)(11,20,23,21)(12,26,27,34)(14,19,15,29)(16,24,32,25)(17,30,33,28)(22,31,38,37)(35,40,36,39);(9,11,13,23)(10,16,18,32)(12,22,27,38)(14,25,15,24)(17,31,33,37)(19,21,29,20)(26,35,34,36)(28,40,30,39);(1,3,5,4)(2,7,6,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,29)(16,24)(17,28)(20,23)(22,37)(25,32)(27,34)(30,33)(31,38)(35,40)(36,39);(1,4,5,3)(2,8,6,7)(9,12,13,27)(10,17,18,33)(11,22,23,38)(14,28,15,30)(16,31,32,37)(19,34,29,26)(20,35,21,36)(24,39,25,40)

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485235);