#507545

order	512
nilpotency class	4
isoclinism	isoclinic class of #500887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
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

535 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)(11,16)(12,24)(13,18)(14,31)(15,20)(17,27)(19,30)(21,38)(22,33)(23,34)(25,36)(26,40)(28,32)(29,35)(37,39);(1,3,4,8)(2,5,6,7)(9,11,13,22,15,23,30,14)(10,16,18,33,20,34,19,31)(12,25,28,27,29,39,26,21)(17,35,37,40,38,24,36,32);(9,12,15,29)(10,17)(11,21,23,27)(13,26,30,28)(14,25,22,39)(16,32)(18,36)(19,37)(20,38)(24,33)(31,35)(34,40);(2,6)(5,7)(9,13,15,30)(10,18,20,19)(11,22,23,14)(12,26,29,28)(16,33,34,31)(17,36,38,37)(21,39,27,25)(24,40,35,32)

Polycyclic presentation:

2^9|b^2=eg,c^2=gh,d^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=fhi,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, 414741);