#7560723

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,23)(13,26)(14,18)(15,32)(16,22)(17,30)(19,31)(21,37)(24,34)(25,36)(27,38)(28,33)(29,40)(35,39);(2,5)(6,8)(9,11,14,22)(10,16,18,20)(12,15,25,31)(13,27,29,21)(17,33,35,34)(19,36,32,23)(24,39,28,30)(26,37,40,38);(1,3,4,7)(2,6,5,8)(9,12,15,22,14,25,31,11)(10,16,19,23,18,20,32,36)(13,28,30,21,29,24,39,27)(17,34,26,37,35,33,40,38);(9,13,15,30,14,29,31,39)(10,17,19,26,18,35,32,40)(11,21,12,24,22,27,25,28)(16,33,23,38,20,34,36,37);(2,5)(6,8)(10,18)(16,20)(17,35)(19,32)(23,36)(26,40)(33,34)(37,38)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=g,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,d^c=di,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467919);