#503563

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

747 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,d^c=di

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, 410759);