#6596183

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
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	#62991	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2917 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6503379);