#6357730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1913 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)(3,13)(4,10)(5,11)(6,14)(7,12)(8,9)(15,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3,5,10,7,15,6,9)(2,8,14,16,12,4,11,13)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,4)(2,9)(3,14)(5,13)(6,16)(7,8)(10,12)(11,15)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,4)(2,10)(3,11)(5,13)(6,16)(7,8)(9,12)(14,15)(18,23)(22,29)(24,30)(28,32);(1,5,7,6)(2,11,12,14)(3,10,15,9)(4,16,8,13)

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264926);