#503951

order	512
nilpotency class	4
isoclinism	isoclinic class of #503866
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24784	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 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)(11,20)(14,25)(18,34)(21,36)(22,30)(24,28)(26,39)(35,38);(1,3,4,8)(2,5,6,7)(9,10,12,17,13,19,15,29)(11,21,30,35,20,36,22,38)(14,26,24,34,25,39,28,18)(16,27,32,23,33,40,31,37);(9,11,13,20)(10,16)(12,22,15,30)(14,27,25,40)(17,31)(18,35)(19,33)(21,26)(23,24,37,28)(29,32)(34,38)(36,39);(2,6)(5,7)(9,12,13,15)(10,17,19,29)(11,22,20,30)(14,24,25,28)(16,31,33,32)(18,26,34,39)(21,38,36,35)(23,27,37,40)

Polycyclic presentation:

2^9|b^2=efh,c^2=f,d^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411147);