#6682637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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

2151 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)(9,10,15,20)(11,22,25,17)(12,26,23,31)(13,21,16,18)(14,19,29,35)(24,37,39,32)(27,40,38,34)(28,36,30,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,16)(13,23)(14,27)(15,25)(18,31)(19,34)(20,22)(21,26)(24,28)(29,38)(30,39)(32,36)(33,37)(35,40);(9,12,13,11,15,23,16,25)(10,17,18,31,20,22,21,26)(14,27,28,24,29,38,30,39)(19,32,33,34,35,37,36,40);(2,6)(5,7)(9,13,15,16)(10,18,20,21)(11,23,25,12)(14,28,29,30)(17,31,22,26)(19,33,35,36)(24,38,39,27)(32,34,37,40);(3,8)(5,7)(9,14)(10,19)(11,24)(12,27)(13,28)(15,29)(16,30)(17,32)(18,33)(20,35)(21,36)(22,37)(23,38)(25,39)(26,40)(31,34)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589833);