#6577605

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
prime radical	#23879	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1951 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,28)(13,19)(14,20)(15,22)(16,23)(18,30)(21,33)(24,36)(25,34)(26,39)(27,35)(29,38)(31,37)(32,40);(1,3,5,4)(2,6,8,7)(9,11,16,13)(10,17,23,19)(12,29,31,14)(15,26,33,27)(18,36,34,40)(20,28,38,37)(21,35,22,39)(24,25,32,30);(1,3,5,4)(2,7,8,6)(9,12)(10,18)(11,24)(13,32)(14,27)(15,25)(16,31)(17,20)(19,38)(21,37)(22,28)(23,34)(26,29)(30,33)(35,40)(36,39);(1,4,5,3)(2,6,8,7)(9,13,16,11)(10,19,23,17)(12,29,31,14)(15,27,33,26)(18,36,34,40)(20,28,38,37)(21,39,22,35)(24,25,32,30);(9,14,16,29)(10,20,23,38)(11,25,13,30)(12,27,31,26)(15,32,33,24)(17,34,19,18)(21,36,22,40)(28,35,37,39)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fi,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, 6484801);