#505411

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

477 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412607);