#504212

order	512
nilpotency class	4
isoclinism	isoclinic class of #504125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=fh,d^2=i,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=gh,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, 411408);