#7565068

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	#24231	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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)(14,30)(15,20)(17,22)(18,28)(19,24)(21,35)(25,31)(26,32)(29,40)(33,39)(34,38)(36,37);(2,6)(7,8)(9,11,15,22)(10,16,20,17)(12,14,26,24)(13,21,29,36)(18,33,34,31)(19,23,30,32)(25,28,39,38)(27,35,40,37);(1,3,5,4)(2,7,6,8)(9,12,24,11,15,26,14,22)(10,17,30,32,20,16,19,23)(13,25,38,21,29,39,28,36)(18,33,40,35,34,31,27,37);(1,3,5,4)(2,7,6,8)(9,13,14,28,15,29,24,38)(10,18,19,27,20,34,30,40)(11,21,12,25,22,36,26,39)(16,31,32,35,17,33,23,37);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fg,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, 7472264);