#6439783

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
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

2042 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,23)(13,18)(15,22)(16,19)(17,27)(20,28)(21,30)(24,26)(25,36)(29,32)(31,34)(33,39)(35,40)(37,38);(2,5)(6,8)(9,11,14,10)(12,24,26,23)(13,29,30,34)(15,22,16,19)(17,28,20,27)(18,31,21,32)(25,35,38,33)(36,39,37,40);(9,12,14,26)(10,17,11,20)(13,25,30,38)(15,28,16,27)(18,33,21,35)(19,24,22,23)(29,37,34,36)(31,40,32,39);(9,13,14,30)(10,18,11,21)(12,25,26,38)(15,31,16,32)(17,33,20,35)(19,29,22,34)(23,36,24,37)(27,39,28,40);(1,3,4,7)(2,6,5,8)(12,26)(17,20)(23,24)(25,38)(27,28)(33,35)(36,37)(39,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346979);