#6351502

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2756 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)(3,11)(4,13)(5,9)(6,10)(7,8)(12,15)(14,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,28)(29,31)(30,32);(1,3)(2,4)(5,12)(6,13)(7,14)(8,15)(9,11)(10,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32);(1,4)(2,3)(5,11)(6,16)(7,15)(8,14)(9,12)(10,13)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,31)(29,32);(2,8)(4,15)(9,10)(11,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,32,31);(1,5,7,6)(2,9,8,10)(3,12,14,13)(4,11,15,16)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,31,32,27)

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258698);