#506224

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=eg,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,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, 413420);