#6348806

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2970 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:

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

Polycyclic presentation:

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

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, 6256002);