#503090

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61251	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

748 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=fh,d^2=e,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, 410286);