#7563836

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	#24171	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 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)(12,23)(13,27)(14,31)(15,20)(17,22)(18,29)(19,24)(21,36)(25,32)(26,33)(28,37)(30,40)(34,39)(35,38);(2,6)(7,8)(9,11,15,22)(10,16,20,17)(12,14,26,24)(13,28,30,21)(18,32,35,34)(19,23,31,33)(25,38,39,29)(27,37,40,36);(1,3,5,4)(2,7,6,8)(9,12,24,11,15,26,14,22)(10,17,31,33,20,16,19,23)(13,25,38,21,30,39,29,28)(18,34,40,36,35,32,27,37);(9,13,14,29,15,30,24,38)(10,18,19,27,20,35,31,40)(11,21,12,25,22,28,26,39)(16,32,33,36,17,34,23,37);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,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, 7471032);