#6585054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
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

1756 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)(11,22)(12,17)(13,19)(14,20)(15,31)(16,21)(18,26)(23,40)(24,34)(25,33)(27,35)(28,36)(29,37)(30,39)(32,38);(2,6)(7,8)(9,11)(10,17)(12,14)(13,27)(15,24)(16,26)(18,31)(19,36)(20,22)(21,34)(23,30)(25,32)(28,29)(33,39)(35,37)(38,40);(9,12,15,26,14,11,16,24)(10,18,21,17,20,34,31,22)(13,25,30,27,29,23,32,28)(19,35,38,33,37,36,39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,14)(13,28)(15,26)(16,24)(18,31)(19,36)(20,22)(21,34)(23,32)(25,30)(27,29)(33,39)(35,37)(38,40);(1,4,5,3)(2,8,6,7)(9,13,14,29)(10,19,20,37)(11,23,12,25)(15,30,16,32)(17,33,22,40)(18,35,34,36)(21,38,31,39)(24,28,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492250);