#6349887

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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)(8,14)(13,16)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,12,10)(17,18,22,26)(19,24,28,32)(20,23,21,25)(27,30,29,31);(1,4,7,3)(2,9,11,12)(5,14,6,13)(8,15,16,10)(17,19,22,28)(18,24,26,32)(20,27,21,29)(23,30,25,31);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,12,16,9)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30);(1,5,7,6)(2,10,11,15)(3,14,4,13)(8,9,16,12)(19,28)(24,32)(27,29)(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,d^b=dh,f^b=fh,g^b=gi,e^c=eh,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, 6257083);