#505109

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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)(11,21)(12,27)(13,17)(14,31)(15,20)(16,24)(18,28)(19,29)(22,40)(23,35)(25,36)(26,39)(30,38)(32,34)(33,37);(1,3,4,8)(2,5,6,7)(9,11,15,33,13,24,30,26)(10,16,20,39,17,21,38,37)(12,23,14,32,28,25,29,22)(18,35,19,34,27,36,31,40);(10,17)(11,22)(12,14)(15,30)(16,34)(18,31)(19,27)(21,40)(23,26)(24,32)(25,33)(28,29)(35,39)(36,37);(2,6)(5,7)(9,12,15,14,13,28,30,29)(10,18,20,19,17,27,38,31)(11,23,33,32,24,25,26,22)(16,35,39,34,21,36,37,40)

Polycyclic presentation:

2^9|b^2=efh,d^2=fhi,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,d^c=dh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412305);