#6576441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

1872 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)(11,18)(12,27)(13,20)(14,21)(15,33)(16,22)(17,23)(19,30)(24,39)(25,35)(26,37)(28,34)(29,36)(31,40)(32,38);(2,5)(6,8)(9,11,17,13)(10,18,23,20)(12,28,31,14)(15,26,16,29)(19,38,35,39)(21,27,34,40)(22,36,33,37)(24,30,32,25);(9,12,17,31)(10,19,23,35)(11,24,13,32)(14,29,28,26)(15,25,16,30)(18,34,20,21)(22,40,33,27)(36,38,37,39);(1,3,4,7)(2,6,5,8)(9,13,17,11)(10,20,23,18)(12,28,31,14)(15,29,16,26)(19,38,35,39)(21,27,34,40)(22,37,33,36)(24,30,32,25);(9,14,17,28)(10,21,23,34)(11,25,13,30)(12,29,31,26)(15,24,16,32)(18,35,20,19)(22,38,33,39)(27,36,40,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,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, 6483637);