#6353503

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353473
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2977 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh

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, 6260699);