#501577

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408773);