#498855

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×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^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, 406051);