#6357737

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

2049 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ghi,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, 6264933);