#7562790

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	#61349	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 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)(9,10,13,15)(11,19,23,34)(12,25,26,39)(14,28,21,18)(16,20,29,24)(17,32,30,27)(22,36,38,31)(33,35,40,37);(2,5)(6,7)(10,15)(11,20)(12,26)(14,21)(16,19)(17,30)(22,37)(23,24)(29,34)(31,33)(35,38)(36,40);(9,11,21,20,13,23,14,24)(10,16,28,34,15,29,18,19)(12,22,32,35,26,38,27,37)(17,31,39,40,30,36,25,33);(9,12,14,27,13,26,21,32)(10,17,18,25,15,30,28,39)(11,22,24,37,23,38,20,35)(16,31,19,33,29,36,34,40);(1,3,4,8)(2,6,5,7)(10,15)(16,29)(17,30)(18,28)(19,34)(25,39)(31,36)(33,40)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,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, 7469986);