#7562789

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	#3035	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 7469985);