#6440845

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60368	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,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, 6348041);