#6583359

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
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	C₂×#135	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490555);