#6575451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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

2106 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)(11,17)(12,19)(13,27)(14,18)(15,21)(16,22)(20,30)(23,33)(24,39)(25,34)(26,37)(28,36)(29,38)(31,35)(32,40);(2,5)(6,8)(9,11,16,25)(10,17,22,34)(12,15,23,14)(13,28,30,26)(18,19,21,33)(20,37,27,36)(24,31,40,29)(32,38,39,35);(10,18)(11,23)(12,25)(14,15)(19,33)(20,35)(21,22)(24,28)(26,40)(27,38)(29,31)(36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,15)(13,24)(14,25)(16,23)(17,21)(18,34)(20,32)(22,33)(26,29)(27,39)(28,31)(30,40)(35,36)(37,38);(9,13,16,30)(10,20,22,27)(11,24,25,40)(12,26,23,28)(14,29,15,31)(17,32,34,39)(18,35,21,38)(19,36,33,37)

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,e^a=ei,g^a=gi,c^b=ch,e^b=eh,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482647);