#6349065

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349016
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	C₄²	C₂⁵
prime radical	C₂×#3497	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3781 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,21,23,32)(12,26,29,34)(13,20,14,19)(16,25,31,24)(17,27,33,28)(22,36,37,30)(35,38,40,39);(2,6)(3,4)(7,8)(9,11)(10,16)(12,22)(13,25)(14,24)(15,23)(17,30)(18,31)(19,32)(20,21)(26,40)(27,39)(28,38)(29,37)(33,36)(34,35);(9,12,15,29)(10,17,18,33)(11,22,23,37)(13,28,14,27)(16,30,31,36)(19,26,20,34)(21,35,32,40)(24,39,25,38);(1,3,5,4)(2,7,8,6)(10,18)(11,23)(17,33)(19,20)(22,37)(24,25)(26,34)(38,39);(2,8)(6,7)(10,18)(16,31)(17,33)(19,20)(21,32)(26,34)(30,36)(35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256261);