#6350339

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

2682 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,e^2=i,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, 6257535);