#7565677

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	#61092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 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:

(1,2,5,6)(3,7,4,8)(9,10,14,18)(11,20,22,16)(12,23,25,35)(13,26,29,40)(15,31,28,19)(17,33,34,30)(21,37,27,38)(24,36,39,32);(2,6)(7,8)(9,11,14,22)(10,16,18,20)(12,15,25,28)(13,27,29,21)(17,32,34,36)(19,35,31,23)(24,33,39,30)(26,37,40,38);(1,3,5,4)(2,7,6,8)(9,12,15,22,14,25,28,11)(10,16,19,23,18,20,31,35)(13,24,30,27,29,39,33,21)(17,32,26,38,34,36,40,37);(1,3,5,4)(2,7,6,8)(9,13,28,33,14,29,15,30)(10,17,31,40,18,34,19,26)(11,21,25,39,22,27,12,24)(16,32,35,37,20,36,23,38);(1,4,5,3)(2,7,6,8)(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=i,c^2=fgi,d^2=fg,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, 7472873);