#6454808

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62810	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,h^b=hi,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, 6362004);