#6583361

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

1662 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)(11,17)(12,24)(13,19)(14,30)(15,20)(16,21)(18,26)(22,32)(23,33)(25,39)(27,34)(28,36)(29,37)(31,35)(38,40);(1,3,4,8)(2,5,6,7)(9,11,16,12)(10,17,21,24)(13,27,29,38)(14,23,15,26)(18,30,33,20)(19,34,37,40)(22,31,25,28)(32,35,39,36);(9,12,15,23,16,11,14,26)(10,18,30,17,21,33,20,24)(13,25,31,38,29,22,28,27)(19,34,36,32,37,40,35,39);(9,11,16,12)(10,17,21,24)(13,27,29,38)(14,23,15,26)(18,30,33,20)(19,34,37,40)(22,31,25,28)(32,35,39,36);(9,13)(10,19)(11,22)(12,25)(14,28)(15,31)(16,29)(17,32)(18,34)(20,35)(21,37)(23,38)(24,39)(26,27)(30,36)(33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,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, 6490557);