#6583820

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
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	#24799	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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)(3,6,7,8)(9,10,14,12)(11,20,23,16)(13,17,30,27)(15,19,26,18)(21,24,34,25)(22,37,38,33)(28,40,39,35)(29,32,36,31);(2,5)(6,8)(9,11,14,23)(10,16,12,20)(13,28,30,39)(15,24,26,25)(17,35,27,40)(18,34,19,21)(22,31,38,32)(29,37,36,33);(9,10,15,18,14,12,26,19)(11,21,24,16,23,34,25,20)(13,17,31,36,30,27,32,29)(22,35,28,33,38,40,39,37);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,16,23,20)(13,29,30,36)(15,19,26,18)(17,32,27,31)(21,24,34,25)(22,35,38,40)(28,33,39,37);(9,13)(10,17)(11,22)(12,27)(14,30)(15,31)(16,33)(18,36)(19,29)(20,37)(21,35)(23,38)(24,28)(25,39)(26,32)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491016);