#6576430

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

1906 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)(11,17)(12,26)(13,19)(14,31)(15,20)(16,21)(18,23)(22,38)(24,35)(25,36)(27,33)(28,34)(29,39)(30,40)(32,37);(2,5)(6,8)(9,11)(10,17)(12,27)(13,30)(14,24)(15,28)(16,25)(18,37)(19,40)(20,34)(21,36)(22,29)(23,32)(26,33)(31,35)(38,39);(9,12,16,30)(10,18,21,39)(11,22,25,32)(13,28,27,24)(14,29,15,23)(17,33,36,19)(20,40,31,26)(34,37,35,38);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,13)(14,24)(15,28)(16,25)(18,38)(19,26)(20,34)(21,36)(22,23)(27,30)(29,32)(31,35)(33,40)(37,39);(9,13)(10,19)(11,23)(12,28)(14,22)(15,32)(16,27)(17,18)(20,37)(21,33)(24,30)(25,29)(26,34)(31,38)(35,40)(36,39)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=f,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, 6483626);