#6448750

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437478
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3207 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)(11,20)(12,17)(13,29)(14,18)(15,19)(16,23)(21,37)(22,25)(24,32)(26,31)(27,34)(28,36)(30,35)(33,38)(39,40);(9,11,15,25)(10,16,19,32)(12,26,27,39)(13,24,14,23)(17,33,34,37)(18,22,29,20)(21,30,38,28)(31,35,40,36);(2,5)(6,8)(9,12)(10,17)(11,21)(13,28)(14,30)(15,27)(16,31)(18,35)(19,34)(20,37)(22,33)(23,26)(24,39)(25,38)(29,36)(32,40);(1,3,4,7)(2,6,5,8)(9,10)(11,22)(12,17)(13,18)(14,29)(15,19)(16,23)(20,25)(21,33)(24,32)(26,31)(27,34)(28,35)(30,36)(37,38)(39,40);(12,27)(17,34)(21,38)(26,39)(28,30)(31,40)(33,37)(35,36)

Polycyclic presentation:

2^9|b^2=i,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6355946);