#6668413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2588 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)(11,15)(12,23)(13,21)(14,19)(16,18)(17,28)(20,25)(22,32)(24,27)(26,37)(29,30)(31,38)(33,35)(34,40)(36,39);(1,3,4,8)(2,5,6,7)(9,11,15,10)(12,24,27,23)(13,21,16,18)(14,30,32,35)(17,25,20,28)(19,33,22,29)(26,34,31,36)(37,39,38,40);(9,12)(10,17)(11,20)(13,25)(14,31)(15,27)(16,28)(18,24)(19,36)(21,23)(22,34)(26,32)(29,40)(30,38)(33,39)(35,37);(2,6)(5,7)(9,13,15,16)(10,18,11,21)(12,25,27,28)(14,29,32,33)(17,24,20,23)(19,35,22,30)(26,39,31,40)(34,38,36,37);(3,8)(5,7)(9,14,15,32)(10,19,11,22)(12,26,27,31)(13,29,16,33)(17,34,20,36)(18,35,21,30)(23,37,24,38)(25,39,28,40)

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^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,e^c=ei,g^c=gi,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575609);