#6575779

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

2108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|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=ehi,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, 6482975);