#6584815

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

1825 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)(11,16)(12,18)(13,27)(14,19)(15,20)(17,22)(21,30)(23,32)(24,34)(25,36)(26,37)(28,31)(29,35)(33,38)(39,40);(2,6)(7,8)(9,11)(10,16)(12,24)(13,22)(14,28)(15,23)(17,27)(18,34)(19,31)(20,32)(21,25)(26,38)(29,39)(30,36)(33,37)(35,40);(1,3,5,4)(2,7,6,8)(9,11,13,22,15,23,14,28)(10,17,19,16,20,31,27,32)(12,21,25,38,26,39,29,24)(18,33,35,30,37,34,36,40);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,24)(13,22)(14,28)(15,23)(17,27)(18,34)(19,31)(20,32)(21,25)(26,38)(29,39)(30,36)(33,37)(35,40);(9,12,15,26)(10,18,20,37)(11,21,23,39)(13,25,14,29)(16,30,32,40)(17,33,31,34)(19,35,27,36)(22,38,28,24)

Polycyclic presentation:

2^9|c^2=fg,d^2=f,e^2=i,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, 6492011);