#6353459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353301
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
prime radical	#63034	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2637 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,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, 6260655);