#7563463

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
prime radical	C₂×#41	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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)(11,16)(12,25)(13,23)(14,18)(15,20)(17,28)(19,31)(21,36)(22,30)(24,33)(26,40)(27,39)(29,34)(32,38)(35,37);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,26)(13,19)(14,22)(15,24)(17,27)(18,30)(20,33)(21,37)(23,31)(25,40)(28,39)(29,38)(32,34)(35,36);(9,12,14,28,15,29,31,37)(10,17,19,25,20,35,18,34)(11,21,23,38,24,39,22,26)(13,27,16,32,30,36,33,40);(9,13,14,16,15,30,31,33)(10,11,19,23,20,24,18,22)(12,27,28,32,29,36,37,40)(17,21,25,38,35,39,34,26);(1,4,5,3)(2,7,8,6)(9,14,15,31)(10,18,20,19)(11,22,24,23)(12,28,29,37)(13,16,30,33)(17,34,35,25)(21,26,39,38)(27,32,36,40)

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=cgi,d^b=dh,g^b=gi,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, 7470659);