#6352011

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2738 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259207);