#6443738

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
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	#23407	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2248 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)(9,10,16,20)(11,17,24,32)(12,15,19,14)(13,18,30,34)(21,31,38,40)(22,25,23,35)(26,29,36,28)(27,33,37,39);(2,5)(6,8)(9,11,16,24)(10,17,20,32)(12,25,19,35)(13,27,30,37)(14,22,15,23)(18,33,34,39)(21,29,38,28)(26,31,36,40);(9,12,15,20,16,19,14,10)(11,17,23,25,24,32,22,35)(13,26,29,34,30,36,28,18)(21,31,27,39,38,40,37,33);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,21)(12,26)(14,28)(15,29)(16,30)(17,31)(19,36)(20,34)(22,37)(23,27)(24,38)(25,39)(32,40)(33,35);(9,14,16,15)(10,12,20,19)(11,22,24,23)(13,28,30,29)(17,25,32,35)(18,26,34,36)(21,37,38,27)(31,39,40,33)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=i,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,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, 6350934);