#6443737

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	#60454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2242 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)(11,16)(12,26)(13,18)(14,19)(15,21)(17,30)(20,35)(22,33)(23,28)(24,36)(25,37)(27,34)(29,38)(31,40)(32,39);(2,5)(6,8)(9,11,15,25)(10,16,21,37)(12,27,31,18)(13,26,34,40)(14,23,35,24)(17,32,38,33)(19,28,20,36)(22,30,39,29);(9,12,14,30,15,31,35,29)(10,17,20,26,21,38,19,40)(11,22,23,27,25,39,24,18)(13,28,32,16,34,36,33,37);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,17)(12,28)(14,32)(15,34)(16,30)(19,39)(20,22)(21,27)(23,26)(24,40)(25,38)(29,37)(31,36)(33,35);(9,14,15,35)(10,19,21,20)(11,23,25,24)(12,29,31,30)(13,32,34,33)(16,28,37,36)(17,26,38,40)(18,39,27,22)

Polycyclic presentation:

2^9|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, 6350933);