#503559

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 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)(9,10,13,19)(11,22,23,17)(12,26,24,40)(14,30,15,32)(16,35,29,21)(18,34,39,25)(20,27,36,31)(28,37,33,38);(1,3,5,4)(2,6,8,7)(9,11,13,23)(10,17,19,22)(12,27,24,31)(14,28,15,33)(16,34,29,25)(18,21,39,35)(20,40,36,26)(30,38,32,37);(1,3,5,4)(2,7,8,6)(9,12,27,11,16,28,14,25,13,24,31,23,29,33,15,34)(10,18,32,37,21,22,20,26,19,39,30,38,35,17,36,40);(1,4,5,3)(2,7,8,6)(10,19)(17,22)(18,39)(20,36)(21,35)(26,40)(30,32)(37,38)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efh,d^2=ei,f^2=hi,g^2=hi,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=gh,h^b=hi,g^c=gi

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, 410755);