#6351944

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

2699 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 6259140);