#6576947

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2603 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)(3,8)(4,6)(5,7)(9,10)(11,23)(12,28)(13,17)(14,19)(15,20)(16,22)(18,31)(21,33)(24,37)(25,38)(26,35)(27,40)(29,34)(30,39)(32,36);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,17,20,23)(12,29,30,14)(16,26,33,27)(18,36,38,37)(19,39,34,28)(21,35,22,40)(24,25,32,31);(9,12)(10,18)(11,24)(13,32)(14,27)(15,30)(16,25)(17,34)(19,23)(20,38)(21,39)(22,28)(26,29)(31,33)(35,37)(36,40);(1,4,5,3)(2,6,7,8)(9,13,15,11)(10,17,20,23)(12,29,30,14)(16,27,33,26)(18,37,38,36)(19,28,34,39)(21,35,22,40)(24,25,32,31);(9,14,15,29)(10,19,20,34)(11,25,13,31)(12,27,30,26)(16,32,33,24)(17,18,23,38)(21,37,22,36)(28,40,39,35)

Polycyclic presentation:

2^9|b^2=f,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=dh,g^c=gi,h^c=hi,e^d=ehi,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, 6484143);