#6353391

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2684 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)(3,6)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(17,18,24,21)(19,26,31,32)(20,25,30,29)(22,27,23,28);(1,3,6,2)(4,10,14,8)(5,11,7,9)(12,15,16,13)(17,19,24,31)(18,25,21,29)(20,22,30,23)(26,28,32,27);(1,4,6,14)(2,8,3,10)(5,13,7,15)(9,16,11,12)(17,20)(18,26)(19,22)(21,32)(23,31)(24,30)(25,28)(27,29);(1,5,6,7)(2,9,3,11)(4,13,14,15)(8,16,10,12)(17,21,24,18)(19,29,31,25)(20,32,30,26)(22,27,23,28);(17,22,24,23)(18,27,21,28)(19,30,31,20)(25,26,29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=i,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, 6260587);