#6576961

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	#60377	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2580 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)(3,8)(4,6)(5,7)(9,10)(11,18)(12,28)(13,19)(14,20)(15,22)(16,25)(17,31)(21,33)(23,40)(24,36)(26,34)(27,39)(29,38)(30,37)(32,35);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,13)(14,25)(15,26)(17,35)(18,20)(19,37)(21,34)(22,39)(23,31)(24,32)(27,33)(28,38)(29,30)(36,40);(9,12,14,30)(10,17,20,36)(11,23,25,32)(13,27,29,26)(15,24,33,31)(16,19,18,38)(21,37,22,28)(34,40,39,35);(1,4,5,3)(2,6,7,8)(9,11)(10,18)(12,29)(13,30)(14,25)(15,26)(16,20)(17,35)(19,37)(21,39)(22,34)(23,24)(27,33)(28,38)(31,32)(36,40);(9,13,14,29)(10,19,20,38)(11,24,25,31)(12,27,30,26)(15,23,33,32)(16,17,18,36)(21,35,22,40)(28,39,37,34)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,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, 6484157);