#6443974

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 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)(3,8)(4,6)(5,7)(9,10,15,21)(11,18,24,17)(12,25,23,32)(13,19,28,36)(14,20,16,30)(22,33,39,31)(26,40,38,34)(27,35,29,37);(1,3,5,4)(2,6,7,8)(9,11,15,24)(10,17,21,18)(12,16,23,14)(13,26,28,38)(19,34,36,40)(20,32,30,25)(22,27,39,29)(31,35,33,37);(9,12,16,11,15,23,14,24)(10,18,20,25,21,17,30,32)(13,26,29,22,28,38,27,39)(19,33,35,40,36,31,37,34);(1,4,5,3)(2,6,7,8)(9,13,15,28)(10,19,21,36)(11,22,24,39)(12,26,23,38)(14,27,16,29)(17,31,18,33)(20,35,30,37)(25,40,32,34);(9,14,15,16)(10,20,21,30)(11,23,24,12)(13,27,28,29)(17,25,18,32)(19,35,36,37)(22,38,39,26)(31,40,33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6351170);