#6450022

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
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	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3004 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,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, 6357218);