#6583563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
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

2072 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)(3,6)(4,5)(7,8)(9,10,14,12)(11,20,23,17)(13,18,29,25)(15,26,16,19)(21,31,34,24)(22,37,39,33)(27,38,40,35)(28,30,36,32);(2,5)(6,8)(9,11)(10,17)(12,20)(13,27)(14,23)(15,24)(16,31)(18,35)(19,21)(22,30)(25,38)(26,34)(28,37)(29,40)(32,39)(33,36);(9,10,15,19,14,12,16,26)(11,21,24,20,23,34,31,17)(13,18,30,28,29,25,32,36)(22,35,40,37,39,38,27,33);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,20,23,17)(13,28,29,36)(15,26,16,19)(18,30,25,32)(21,24,34,31)(22,38,39,35)(27,37,40,33);(9,13)(10,18)(11,22)(12,25)(14,29)(15,30)(16,32)(17,33)(19,28)(20,37)(21,35)(23,39)(24,40)(26,36)(27,31)(34,38)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^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=dh,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, 6490759);