#7563832

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	#3511	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,25)(13,27)(14,28)(15,20)(17,18)(19,30)(21,38)(22,39)(23,37)(24,32)(26,40)(29,31)(33,35)(34,36);(2,6)(7,8)(9,11,15,24)(10,16,20,32)(12,26,27,39)(13,22,25,40)(14,29,30,23)(17,33,34,38)(18,35,36,21)(19,37,28,31);(1,3,5,4)(2,7,6,8)(9,12,14,18,15,27,30,36)(10,17,19,25,20,34,28,13)(11,21,23,39,24,35,29,26)(16,22,31,38,32,40,37,33);(9,13,14,17,15,25,30,34)(10,18,19,27,20,36,28,12)(11,22,23,38,24,40,29,33)(16,21,31,39,32,35,37,26);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=g,e^2=f,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, 7471028);