#6448082

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355278);