#6574583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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×#578	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1823 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)(11,22,17,16)(12,25,27,24)(13,18,30,37)(15,21,33,20)(23,39,35,34)(26,40,36,28)(29,31,38,32);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,16,19,22)(12,21,27,20)(13,28,30,40)(15,24,33,25)(18,36,37,26)(23,31,35,32)(29,34,38,39);(9,12,14,27)(10,17,19,11)(13,26,30,36)(15,16,33,22)(18,35,37,23)(20,24,21,25)(28,38,40,29)(31,34,32,39);(9,10,14,19)(11,16,17,22)(12,25,27,24)(13,29,30,38)(15,20,33,21)(18,31,37,32)(23,36,35,26)(28,34,40,39);(9,13)(10,18)(11,23)(12,26)(14,30)(15,31)(16,34)(17,35)(19,37)(20,29)(21,38)(22,39)(24,28)(25,40)(27,36)(32,33)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,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=dh,g^c=gi,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, 6481779);