#6583357

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,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, 6490553);