#6348491

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348447
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

2729 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,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, 6255687);