#6350182

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

2856 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=i,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, 6257378);