#507598

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	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

426 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)(11,16)(12,17)(13,19)(14,28)(15,20)(18,26)(21,35)(22,32)(23,30)(24,31)(25,33)(27,34)(29,36)(37,39)(38,40);(2,5)(6,7)(9,11,13,23,15,24,14,21)(10,16,19,30,20,31,28,35)(12,22,25,37,27,38,26,36)(17,32,33,39,34,40,18,29);(10,17,20,34)(11,21)(12,25)(13,14)(16,29,31,39)(18,28,33,19)(23,24)(26,27)(30,40,35,32)(36,37);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=gh,d^2=e,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, 414794);