#6576955

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	#60373	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2647 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)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,30)(24,31)(25,32)(26,34)(27,33)(28,36)(29,35)(37,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11,16,25)(10,17,22,32)(12,14,24,15)(13,26,28,37)(18,31,21,19)(20,34,36,40)(23,29,38,27)(30,35,39,33);(10,18)(11,12)(14,15)(19,31)(20,33)(21,22)(23,26)(24,25)(27,29)(34,40)(35,36)(37,38);(1,4,5,3)(2,6,7,8)(9,12)(10,19)(11,14)(13,23)(15,25)(16,24)(17,18)(20,30)(21,32)(22,31)(26,29)(27,37)(28,38)(33,40)(34,35)(36,39);(9,13,16,28)(10,20,22,36)(11,23,25,38)(12,26,24,37)(14,27,15,29)(17,30,32,39)(18,33,21,35)(19,34,31,40)

Polycyclic presentation:

2^9|b^2=fi,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, 6484151);