#6579426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(9,10,16,23)(11,24,13,20)(12,19,32,38)(14,22,15,21)(17,28,35,27)(18,30,39,31)(25,34,29,40)(26,36,33,37);(2,5)(6,8)(9,11,15,28,16,13,14,27)(10,17,22,20,23,35,21,24)(12,26,31,29,32,33,30,25)(18,36,19,34,39,37,38,40);(1,3,4,7)(2,6,5,8)(9,12,14,30,16,32,15,31)(10,18,21,38,23,39,22,19)(11,25,27,33,13,29,28,26)(17,34,24,36,35,40,20,37);(9,12)(10,19)(11,26)(13,33)(14,30)(15,31)(16,32)(17,34)(18,21)(20,37)(22,39)(23,38)(24,36)(25,27)(28,29)(35,40);(9,13)(10,20)(11,16)(12,29)(14,28)(15,27)(17,21)(18,37)(19,40)(22,35)(23,24)(25,32)(26,30)(31,33)(34,38)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486622);