#498854

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#509	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1638 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)(3,4)(9,10)(11,21)(12,18)(13,17)(14,30)(15,20)(16,23)(19,28)(22,36)(24,40)(25,37)(26,33)(27,35)(29,34)(31,38)(32,39);(2,4)(5,6)(7,8)(9,11,15,25)(10,16,20,32)(12,24,14,22)(13,27,29,38)(17,21,34,37)(18,35,30,31)(19,36,33,40)(23,26,39,28);(6,8)(10,17)(12,14)(13,28)(16,21)(18,19)(20,34)(22,24)(23,27)(26,29)(30,33)(31,40)(32,37)(35,36)(38,39);(9,12,15,14)(10,18,20,30)(11,22,25,24)(13,26,29,28)(16,31,32,35)(17,33,34,19)(21,36,37,40)(23,38,39,27)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406050);