#6349842

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

2821 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)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,18,22,26)(19,24,28,32)(20,23,21,25)(27,30,29,31);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,31)(26,32);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,15,16,14)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30);(3,12)(8,16)(11,13)(14,15)(19,28)(24,32)(27,29)(30,31)

Polycyclic presentation:

2^9|b^2=i,d^2=h,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, 6257038);