#6350216

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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

2743 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257412);