#6583819

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	#61337	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)(11,21,24,16)(13,18,30,27)(15,20,17,19)(22,26,35,25)(23,37,39,33)(28,38,40,34)(29,31,36,32);(2,5)(6,8)(9,11)(10,16)(12,21)(13,28)(14,24)(15,25)(17,26)(18,34)(19,22)(20,35)(23,32)(27,38)(29,33)(30,40)(31,39)(36,37);(9,12,15,20,14,10,17,19)(11,22,25,21,24,35,26,16)(13,27,31,36,30,18,32,29)(23,38,28,37,39,34,40,33);(1,3,4,7)(2,6,5,8)(9,10,14,12)(11,21,24,16)(13,29,30,36)(15,19,17,20)(18,31,27,32)(22,25,35,26)(23,38,39,34)(28,37,40,33);(9,13)(10,18)(11,23)(12,27)(14,30)(15,31)(16,33)(17,32)(19,29)(20,36)(21,37)(22,38)(24,39)(25,28)(26,40)(34,35)

Polycyclic presentation:

2^9|a^2=f,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, 6491015);