#6357908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
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

1897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=di,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, 6265104);