#6575787

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

2062 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)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,31)(24,30)(25,32)(26,34)(27,33)(28,36)(29,35)(37,39)(38,40);(2,5)(6,8)(9,11,16,25)(10,17,22,32)(12,15,23,14)(13,26,28,37)(18,19,21,31)(20,34,36,39)(24,27,38,29)(30,33,40,35);(10,18)(11,23)(12,25)(14,15)(19,31)(20,33)(21,22)(24,37)(26,38)(27,29)(34,39)(35,36);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,15)(13,24)(14,25)(16,23)(17,21)(18,32)(20,30)(22,31)(26,27)(28,38)(29,37)(33,34)(35,39)(36,40);(9,13,16,28)(10,20,22,36)(11,24,25,38)(12,26,23,37)(14,27,15,29)(17,30,32,40)(18,33,21,35)(19,34,31,39)

Polycyclic presentation:

2^9|b^2=i,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, 6482983);