#6576091

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×#577	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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)(11,24)(12,30)(13,19)(14,20)(15,21)(16,23)(17,28)(18,32)(22,26)(25,39)(27,38)(29,36)(31,35)(33,40)(34,37);(2,5)(6,8)(9,11)(10,17)(12,31)(13,26)(14,34)(15,28)(16,29)(18,20)(19,23)(21,24)(22,36)(25,32)(27,33)(30,40)(35,38)(37,39);(9,12,15,27)(10,18,21,37)(11,25,28,14)(13,33,29,31)(16,34,26,32)(17,35,24,40)(19,20,36,39)(22,38,23,30);(1,3,4,7)(2,6,5,8)(9,13,15,29)(10,19,21,36)(11,26,28,16)(12,14,27,25)(17,23,24,22)(18,35,37,40)(20,38,39,30)(31,34,33,32);(9,14,15,25)(10,20,21,39)(11,27,28,12)(13,32,29,34)(16,31,26,33)(17,30,24,38)(18,36,37,19)(22,40,23,35)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^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, 6483287);