#6353538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353473
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

2781 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)(3,11)(4,13)(5,8)(6,9)(7,10)(12,15)(14,16)(17,18,23,20)(19,26,29,32)(21,25,22,30)(24,31,27,28);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,19)(18,24)(20,27)(21,28)(22,31)(23,29)(25,32)(26,30);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,18,23,20)(19,24,29,27)(21,25,22,30)(26,28,32,31);(1,2)(3,4)(5,9)(6,8)(7,10)(11,13)(12,16)(14,15)(17,20,23,18)(19,27,29,24)(21,25,22,30)(26,28,32,31);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)

Polycyclic presentation:

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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260734);