#6351898

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

2867 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)(3,10)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(20,29)(24,28)(25,32)(27,30);(1,2,7,4)(3,8,13,11)(5,9,6,14)(10,12,16,15)(17,19)(18,22)(20,30)(21,26)(23,31)(24,32)(25,28)(27,29);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(18,23)(22,31)(24,28)(25,32);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,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, 6259094);