#6595651

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

3088 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:

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

Polycyclic presentation:

2^9|a^2=i,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,f^a=fi,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, 6502847);