#6452230

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3245 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,16)(13,19)(14,20)(15,22)(18,29)(21,33)(23,31)(24,36)(25,30)(26,34)(27,37)(28,35)(32,38)(39,40);(9,11,13,22,14,24,29,23)(10,15,18,17,20,31,19,36)(12,25,27,38,26,39,28,21)(16,32,35,30,34,33,37,40);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,21)(13,27)(14,26)(15,30)(17,33)(18,35)(19,37)(20,34)(22,25)(23,39)(24,38)(28,29)(31,40)(32,36);(1,4,5,3)(2,6,7,8)(9,11)(10,17)(12,21)(13,22)(14,24)(15,19)(16,33)(18,31)(20,36)(23,29)(25,27)(26,38)(28,39)(30,37)(32,34)(35,40);(12,26)(16,34)(21,38)(25,39)(27,28)(30,40)(32,33)(35,37)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6359426);