#7563456

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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	C₂×AGL(1,ℤ/8)	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470652);