#6576431

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,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=ehi,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,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, 6483627);