#502376

order	512
nilpotency class	4
isoclinism	isoclinic class of #502081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=i,d^2=ei,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=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, 409572);