#503718

order	512
nilpotency class	4
isoclinism	isoclinic class of #503654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

449 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)(4,5)(6,8)(9,10,13,19,12,18,15,17)(11,21,24,32,23,16,29,31)(14,20,27,35,26,34,28,33)(22,38,40,30,39,36,25,37);(2,5)(6,7)(9,11)(10,16)(12,23)(13,24)(14,22)(15,29)(17,32)(18,21)(19,31)(20,36)(25,28)(26,39)(27,40)(30,33)(34,38)(35,37);(10,17)(11,22,23,39)(13,15)(14,27)(16,30,21,37)(18,19)(24,25,29,40)(26,28)(31,38,32,36)(33,35);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,19)(20,34)(30,37)(31,32)(33,35)(36,38)

Polycyclic presentation:

2^9|a^2=f,c^2=fh,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410914);