#6349354

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349298
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6256550);