#7566054

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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	#61094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1901 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:

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

Polycyclic presentation:

2^9|a^2=f,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=cg,d^b=dh,g^b=gi,e^c=ei,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, 7473250);